Linear Algebra - Distance,Hyperplanes and Halfspaces,Eigenvalues,Eigenvectors ( Continued 3 )

Summary

This lecture concludes a series on linear algebra for data science, focusing on the relationships between eigenvectors and fundamental subspaces. The instructor explains the significance of symmetric matrices, highlighting that they always have real eigenvalues and linearly independent eigenvectors. These concepts are crucial in data science, particularly for covariance matrices and algorithms like principal component analysis (PCA). The lecture connects eigenvectors to null space and column space, providing foundational knowledge for further study in regression analysis and machine learning.

Takeaways

• 🧮 Symmetric matrices are frequently used in data science, especially in algorithms and covariance matrices.
• 🔢 The eigenvalues of symmetric matrices are always real, and the corresponding eigenvectors are also real.
• ♻️ For symmetric matrices, we are guaranteed to have n linearly independent eigenvectors, even if some eigenvalues are repeated.
• 🔗 Eigenvectors corresponding to zero eigenvalues are found in the null space of the matrix, while those corresponding to non-zero eigenvalues span the column space.
• 🚫 If a matrix is full rank (none of the eigenvalues are zero), there will be no vectors in the null space.
• 🧩 The eigenvectors of symmetric matrices that correspond to non-zero eigenvalues form a basis for the column space.
• 📐 The connection between eigenvectors, null space, and column space is important for data science algorithms like principal component analysis (PCA).
• 🔍 Eigenvectors of symmetric matrices are linear combinations of the matrix's columns.
• 📊 Symmetric matrices of the form A^T A or A A^T are frequently encountered in data science computations and always have non-negative eigenvalues.
• 📚 The lecture series covers essential linear algebra concepts for data science, laying the foundation for further topics in regression analysis and machine learning.

Q & A

What happens when a matrix is symmetric?

When a matrix is symmetric, its eigenvalues are always real, and it guarantees that there are n linearly independent eigenvectors, even if eigenvalues are repeated.

Why are symmetric matrices important in data science?

Symmetric matrices are important in data science because they frequently occur in computations, such as the covariance matrix, and they have useful properties like real eigenvalues and guaranteed linearly independent eigenvectors.

What is the significance of eigenvalues being real for symmetric matrices?

For symmetric matrices, real eigenvalues imply that the corresponding eigenvectors are also real, making the matrix easier to work with in practical applications like data science and machine learning.

How are eigenvectors related to the null space when the eigenvalue is zero?

Eigenvectors corresponding to eigenvalue zero are in the null space of the matrix. If an eigenvalue is zero, the corresponding eigenvector lies in the null space.

What is the connection between eigenvectors and the column space for symmetric matrices?

For symmetric matrices, the eigenvectors corresponding to nonzero eigenvalues form a basis for the column space. This means that the column space can be described using these eigenvectors.

What role do repeated eigenvalues play in the context of eigenvectors?

When eigenvalues are repeated, there may be fewer linearly independent eigenvectors for a general matrix. However, for symmetric matrices, even with repeated eigenvalues, there will still be n linearly independent eigenvectors.

How do a transpose A and A transpose matrices relate to symmetric matrices in data science?

Both A transpose A and A A transpose are symmetric matrices, which frequently occur in data science computations, such as covariance matrices. Their symmetry guarantees real, non-negative eigenvalues and linearly independent eigenvectors.

What does it mean if a matrix has no eigenvalues equal to zero?

If a matrix has no eigenvalues equal to zero, it is full rank, meaning there are no vectors in the null space. This implies that all eigenvectors are outside the null space.

How are eigenvectors computed for a symmetric matrix with repeated eigenvalues?

For a symmetric matrix with repeated eigenvalues, the eigenvectors can still be computed to be linearly independent, ensuring that the matrix has the full set of n independent eigenvectors.

What is the importance of the relationship between eigenvalues, null space, and column space in linear algebra?

The relationship between eigenvalues, null space, and column space is critical in linear algebra because it helps define the structure of a matrix. Eigenvectors corresponding to zero eigenvalues belong to the null space, while eigenvectors corresponding to nonzero eigenvalues define the column space. These concepts are foundational in data science and machine learning algorithms like PCA.

Outlines

00:00

📊 The Role of Eigenvalues and Eigenvectors in Data Science

This section discusses the importance of eigenvalues and eigenvectors in data science, specifically how they relate to linear algebra concepts. The eigenvalue equation (A - λI = 0) is explained, emphasizing how eigenvalues can be real or complex. However, for symmetric matrices, eigenvalues and their corresponding eigenvectors are always real. Symmetric matrices, like the covariance matrix, play a crucial role in data science. This section also notes that for matrices with distinct eigenvalues, there are guaranteed to be linearly independent eigenvectors, but this is not always true for repeated eigenvalues, unless the matrix is symmetric.

05:01

🔄 Symmetric Matrices and their Special Properties

This paragraph delves deeper into the properties of symmetric matrices, which have real and non-negative eigenvalues. These matrices appear frequently in data science computations, such as AᵀA or AAᵀ. Because these matrices are symmetric, they guarantee the existence of n linearly independent eigenvectors. The relationship between these eigenvectors and the matrix's column and null spaces is introduced, providing a foundation for understanding eigenvalues’ role in determining matrix rank and space interactions.

10:05

🧮 Eigenvalues, Null Spaces, and Full-Rank Matrices

Here, the focus shifts to the relationship between eigenvectors and the null space of a matrix. If an eigenvalue is zero, its corresponding eigenvector lies in the null space. Conversely, eigenvectors with non-zero eigenvalues cannot belong to the null space. This section explains how a matrix with no zero eigenvalues is full rank, implying no vectors exist in its null space. The connection between eigenvalues and the full rank of a matrix is a key point in this analysis.

15:07

📐 Column Space and Eigenvectors of Symmetric Matrices

This paragraph elaborates on the relationship between eigenvectors and the column space in symmetric matrices. If a matrix has r zero eigenvalues, the corresponding eigenvectors span the null space, while the remaining n-r eigenvectors span the column space. Through the rank-nullity theorem, the text explains that the rank of the matrix is n-r, meaning there are n-r independent column vectors. The eigenvectors corresponding to non-zero eigenvalues form a basis for the matrix's column space.

20:08

🔗 Linear Combinations of Eigenvectors and Column Space

In this section, the text explores how eigenvectors corresponding to non-zero eigenvalues are linear combinations of the matrix's columns. It shows that these eigenvectors form a basis for the column space, and each eigenvector can be expressed as a weighted combination of the matrix’s columns. The paragraph demonstrates how these eigenvectors span the column space and help reduce the complexity of the matrix representation by focusing on the independent vectors.

📝 Example of Eigenvalues and Eigenvectors in Action

An example of a 3x3 symmetric matrix is presented to illustrate the theoretical concepts discussed earlier. The example matrix has eigenvalues of 0, 1, and 2, and corresponding eigenvectors are calculated. The vector corresponding to the zero eigenvalue lies in the null space, while the others span the column space. The text verifies these relationships using matrix multiplication, demonstrating how the eigenvectors represent relationships between the matrix’s variables and span the column space.

📚 Conclusion: Key Takeaways on Symmetric Matrices

The final section summarizes the lecture series on linear algebra for data science. It revisits the critical points, such as the fact that symmetric matrices always have real eigenvalues and n linearly independent eigenvectors. The connection between eigenvectors and the null space and column space of a matrix is reiterated, particularly in symmetric matrices. The importance of these concepts in algorithms like principal component analysis (PCA) is highlighted, as well as their broader applications in data science. The lecture concludes by mentioning upcoming modules on statistics and machine learning.

Keywords

💡Eigenvalue

An eigenvalue is a scalar that arises in the context of the eigenvalue equation, which is central to linear algebra. In the video, the eigenvalue is part of the equation A*x = λ*x, where A is a matrix, λ is the eigenvalue, and x is the corresponding eigenvector. Eigenvalues can be real or complex, and the script emphasizes their importance in data science, especially in relation to symmetric matrices where eigenvalues are always real.

💡Eigenvector

An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it. In the context of the video, eigenvectors are paired with eigenvalues and are key in determining the behavior of matrices in linear algebra. The video explains that for symmetric matrices, eigenvectors are always real and linearly independent, and they play a significant role in defining the column space and null space of matrices.

💡Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose, meaning that the elements are mirrored along the diagonal. In the video, symmetric matrices are highlighted for their special properties, such as always having real eigenvalues and guaranteeing linearly independent eigenvectors. The covariance matrix, commonly used in data science, is an example of a symmetric matrix.

💡Null Space

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector. The video discusses the relationship between the null space and eigenvectors, particularly how eigenvectors corresponding to a zero eigenvalue are part of the null space. This concept is important for understanding the structure of matrices, especially in relation to their rank.

💡Column Space

The column space of a matrix is the set of all possible linear combinations of its column vectors. In the video, the connection between eigenvectors and the column space is explored, showing that for symmetric matrices, the eigenvectors corresponding to nonzero eigenvalues form a basis for the column space. This is a crucial concept in data science for understanding matrix transformations.

💡Rank

The rank of a matrix is the dimension of its column space, representing the number of linearly independent columns. In the video, rank is discussed in the context of full-rank matrices, where the null space only contains the zero vector. The rank-nullity theorem, which links the rank and null space of a matrix, is also mentioned as a key result in linear algebra.

💡Covariance Matrix

A covariance matrix is a symmetric matrix that represents the covariance between different variables in a dataset. In data science, it is often used in algorithms like Principal Component Analysis (PCA). The video references the covariance matrix as an example of a symmetric matrix with real eigenvalues, highlighting its importance in data analysis.

💡Full Rank Matrix

A full-rank matrix is one where the rank is equal to the number of its columns, implying that all columns are linearly independent. In the video, it is mentioned that when a matrix is full-rank, its null space only contains the zero vector. This property is important in ensuring that systems of linear equations have unique solutions.

💡Principal Component Analysis (PCA)

Principal Component Analysis is a dimensionality reduction technique used in data science that relies on the eigenvectors and eigenvalues of the covariance matrix to identify the directions of maximum variance in the data. The video touches on PCA as an application of the concepts of eigenvalues and eigenvectors, showing how these linear algebra principles are foundational in data science algorithms.

💡Rank-Nullity Theorem

The rank-nullity theorem is a fundamental theorem in linear algebra that states that the rank of a matrix plus the dimension of its null space equals the number of its columns. In the video, this theorem is used to explain the relationship between the rank of a matrix and the number of independent eigenvectors corresponding to nonzero eigenvalues, particularly in symmetric matrices.

Highlights

Introduction to the connection between eigenvectors and fundamental subspaces in linear algebra for data science.

Eigenvalue-eigenvector equation: A - λI = 0, showcasing the polynomial nature and potential for real or complex eigenvalues in general matrices.

Symmetric matrices have special properties in data science, including real eigenvalues and guaranteed real eigenvectors.

Symmetric matrices are common in data science, e.g., covariance matrices, and ensure n linearly independent eigenvectors.

For symmetric matrices, repeated eigenvalues still guarantee independent eigenvectors, unlike general matrices.

The identity matrix serves as an example of a symmetric matrix with repeated eigenvalues but independent eigenvectors.

Matrices of the form AᵀA or AAᵀ, commonly encountered in data science, are always symmetric and have non-negative eigenvalues.

Eigenvectors corresponding to zero eigenvalues are in the null space of matrix A, establishing a key relationship.

When none of the eigenvalues are zero, the matrix is full rank, indicating no non-trivial solutions to Ax = 0.

For symmetric matrices, eigenvectors corresponding to non-zero eigenvalues span the column space of the matrix.

The rank-nullity theorem links the number of non-zero eigenvalues to the rank and nullity of the matrix.

Eigenvectors are linear combinations of the columns of matrix A, and for symmetric matrices, these eigenvectors form a basis for the column space.

In symmetric matrices, the connection between null space, column space, and eigenvectors forms the foundation of data science algorithms.

Practical example: A symmetric 3x3 matrix is used to compute eigenvalues and eigenvectors, verifying theoretical results.

Eigenvectors corresponding to zero eigenvalues can identify relationships between variables in data science problems.

Transcripts

[Music]

this is the last lecture in the series

of lectures on linear algebra for data

science and as I mentioned in the last

class today I'm going to talk to you

about the connections between

eigenvectors on the fundamental

subspaces that we have described earlier

we saw in the last lecture that the

eigenvalue eigenvector equation this

else in this equation having to be

satisfied which is a minus lambda I

equals 0 in general

we also saw that this would turn out to

be a polynomial of degree n in lambda

which basically means that even if this

matrix a is real because the solutions

to a polynomial equation could be either

real or complex you could have

eigenvalues that are complex so for a

general matrix

you could have eigenvalues which are

either real or complex and notice that

since we write the equation ax equals

lambda X whenever this eigen values

become complex then the eigen vectors

are also complex vectors so this is true

in general however if the matrix is

symmetric and symmetric matrices are of

the form equal to a transpose then there

are certain nice properties for these

matrices which are very useful for us in

data science we also encounter symmetric

matrices quite a bit in data science for

example the covariance matrix turns out

to be a symmetric matrix and there are

several other cases where we deal with

symmetric matrices so these properties

of symmetric matrices are very useful

for us when we look at algorithms in

data science now the first property of

symmetric matrices that is very useful

to us is if the matrix is symmetric then

the eigenvalues are always real so

irrespective of what the symmetric

matrix is this polynomial would

give real solutions for symmetric

matrices and as I mentioned before if

this turns out to be real then the

eigenvectors are also real now there is

another aspect of an eigen values and

eigenvectors that is important if I have

a matrix a and I have n different

eigenvalues lambda 1 to lambda and all

of them are distinct then I'll

definitely have n linearly independent

eigenvectors corresponding to them which

could be nu 1 nu 2 all the way up to nu

n however if there are certain

eigenvalues which are repeated so for

example if you take a case where I can

value lambda 1 is repeated then I could

have some polynomial which is like this

so the polynomial original polynomial

has eigen value lambda 1 repeated twice

and then there's another n minus tooth

order polynomial which will give you n

minus 2 other solutions now in this case

when I have lambda 1 repeated like this

then it could turn out that this eigen

value either has to I can victors which

are independent or it might have just

one eigenvector so finding n linearly

independent eigenvectors is not always

guaranteed for any general matrix and we

already know that eigenvectors could be

complex for any general matrix however

when we talk about symmetric matrices we

can say for sure that the eigenvalues

would be real the eigenvectors would be

real further we are always guaranteed

that we'll have n linearly independent

eigenvectors for symmetric matrices it

doesn't matter how many times the

eigenvalues get repeated one classic

example of a symmetric matrix where

eigen values are repeated many times so

take identity matrix something like this

here

this identity matrix has eigen value

lambda equal to one which is repeated

thrice but it would have three

independent eigen vectors 1 0 0 0 1 0

and 0 0 so this is a case where I can

values repeated tries but there are

three independent eigenvectors so this

is also an important result that we

should keep in mind and as I mentioned

in the last slide symmetric matrices

have a very important role in data

sciences in fact symmetric matrices of

the type a transpose a or EA a transpose

are often encountered in data sense

computations and notice that both of

these matrices are symmetric so for

example if I take a transpose a

transpose this will be a transpose a

transpose transpose which will be a

transpose a so the transpose of the

matrix is the same you can verify that a

a transpose is also symmetric through

the same idea so we know matrices of the

form a transpose a or a a a transpose

are both symmetric and they are often

encountered when we do computations in

data science and we know from the

previous slide I had mentioned that the

eigenvalues of symmetric matrices are

real

if the symmetric matrix also takes this

form or this form we can also say that

while the eigenvalues are real they are

also non-negative that is they can they

will be either 0 or positive but none of

the eigenvalues will be negative so this

is another important idea that we will

use when we do data science when we look

at covariance matrices and so on also

the fact that this a transpose a and a

transpose are symmetric matrices

guarantees that there will be n linearly

independent eigenvectors for matrices of

this form also so what we are going to

do right now is because of the

importance of symmetric matrices in data

sense computations we are going to look

at the connection between the

eigenvectors and the column space a null

space

for a symmetric matrix some of these

results translate to non symmetric

matrices also but for symmetric matrices

all of these are results that we can use

so we go back to the eigenvalue

eigenvector equation a new is lambda nu

and this result that we are going to

talk about right now it's true whether

the matrix a is symmetric or not

if a nu equals lambda nu we ask the

question what happens when lambda is

zero that is one of the eigenvalues

becomes zero so when one of the

eigenvalues becomes zero then we have

this equation which is a nu equals zero

so we can interpret new as an

eigenvector corresponding to eigenvalue

zero we have also seen this equation

before when we talked about different

subspaces for matrices

we saw that null space vectors are of

the form a beta is zero from one of our

initial lectures you notice that this

and this form are the same so that

basically means that nu which is an

eigenvector corresponding like

corresponding to eigenvalue lambda

equals 0 is a null space vector because

it is just of the form that we have here

so we could say the eigenvectors

corresponding to 0 eigen values are in

the null space of the original matrix a

conversely if the eigenvalue

corresponding to an eigen vector is not

0 then that eigenvector cannot be in the

null space of a so these are important

results that we need to know so this is

how heigen vectors are connected to null

space if none of the eigen values are 0

that basically means that the matrix a

is full rank and that means that I can

never solve

a new equal to 0 and get non-trivial new

so it's not possible if a is full right

so if a is full blank I cannot solve for

this and get non-trivial new so whenever

lambda is lambda such that there are

there is no eigen value that is 0 that

means a is full rank matrix that means

there is no eigen vector such that a new

is 0 which basically means that there

are no vectors in the null space now

let's see the connection between

eigenvectors and column space in this

case i'm going to show you the result

and this result is valid for symmetric

matrices let us assume that i have a

symmetric matrix a and the symmetric

matrix a we know will have n real

eigenvalues let's assume that all of

these eigen values are 0 so this are

could be 0 also that means there is no

eigenvalue which is 0 so even then all

of this discussion is valid but as a

general case let us assume that our i

gain values of 0 so there are our 0

eigen values and since we are assuming

this matrix is n by n there will be n

real eigen values of which are are 0 so

there will be n minus R nonzero eigen

values and from the previous slide we

know that the our eigen vectors

corresponding to this are 0 eigen values

are all in the null space okay so since

I have all 0 eigen values I will have

our eigen vectors corresponding to this

so all of these are eigen vectors are in

the null space which basically means

that the dimension of the null space is

are because there are our vectors in the

null space and from rank nullity theorem

we know that rank plus nullity is equal

to number of columns in this case n

since there are our eigen vectors in the

null space nullities are so the rank of

the matrix has to be equal to n minus r

so that is what we are saying here and

further we know that column rank is

equal to row rank and since the rank of

the matrix is n minus R the column rank

also has to be n minus R this basically

means that there are n minus R

independent vectors in the columns of

the matrix so one question that we might

ask is the following we could ask what

could be a basis set for this column

space or what could be the n minus r

independent vectors that we can use as

the columns subspace so there are a few

things that we can notice based on what

we have discussed till now first notice

that the n minus r eigenvectors that we

talked about in the last slide the ones

that are not i ghen vectors

corresponding to lambda equal to 0 they

cannot be in the null space because

lambda is a number which is different

from 0 so these n minus r eigenvectors

cannot be in the null space of the

matrix a so let me write again we are

discussing all of this for symmetric

matrices we know that all of this n

minus I are Reagan vectors are also

independent because we said irrespective

of what the symmetric matrix is we will

always get n linearly independent eigen

vectors so that means these n minus r

eigenvectors are also independent we

also know that each of these independent

eigenvectors are going to be linear

combinations of columns of a to see this

let us look at this equation so let me

write this out so I could call this as a

I am going to expand this nu into

new one new - all the way up to new

again notice that these are components

of new we are just taking one

eigenvector new and then these are the

end components in that eigenvector I can

write this as lambda mu and from the

previous lecture of how to do this

column multiplication and how to

interpret this column multiplication I

said you could think of this as nu 1

times the first column of a + nu 2 times

the second column of a all the way up to

Nu n types nth column of a equal to

lambda nu now in this equation let me be

very clear

these are scalars which are components

in the eigenvector nu these are column

vectors is the first column of a second

column of a this is n column of a this

is again a scalar lambda which is the

eigen value corresponding to Nu so this

could be true for any of the N minus R

eigen vectors which are not in the null

space of this matrix a now take lambda

to the other side so you will have this

equation as nu is nu 1 by lambda a 1 and

so on plus nu n by lambda a again new

one is a scalar lambda is a scalar so

these are all constants that we are

using to multiply these columns now you

will clearly see that each of this eigen

vectors n minus R eigen vectors are

linear combinations of the columns of a

so there are n minus R linear in

linearly independent eigen vectors like

this and each of this are combinations

of columns of a and we also know that

the dimension of the column space is n

minus R in other words if you take all

of this columns a 1 to a n these can be

represented using just n minus R

linearly independent vectors now when we

put all of these facts together which is

the N minus R eigen vectors are linearly

independent

there are combinations of columns of a

and the number of independent columns of

a can be only n minus R so this implies

that the eigenvectors corresponding to

the nonzero eigenvalues for a symmetric

matrix form a basis for the column space

so this is the important result that I

wanted to show you with all of these

ideas now again these results we will

see and use as we look at some of the

data finds algorithms later so let's

take a simple example to understand how

all of this works let's consider a

matrix which is of this form here it's a

three by three matrix first thing that I

want you to notice that this is a

symmetric matrix so if you do a

transpose equals saying

and we said symmetric matrices will

always have real eigenvalues and when

you do the eigen value computation for

this the way you do the eigen value

computation is you take determinant a

minus lambda I equal to zero then you're

going to get a third order polynomial

you set it equal to zero and then you

calculate the three solutions to this

polynomial and these would turn out to

be the solution 0 1 2 and you take each

of these solutions and then substitute

it back in and then solve for ax equals

lambda X then you would get the three

eigen vectors corresponding to this

which are given by this this and this

notice from our discussion before since

this is an eigen vector corresponding to

lambda equal to 0 so this is going to be

in the null space of this matrix a and

these are the remaining two how do I get

this 2 which is 3 minus 1 n n by n so

it's a 3 by 3 matrix and nullity is 1

because there is only one eigen vector

corresponding to lambda equal to 0 so I

get two other linearly independent

vectors and in the last slide when we

were discussing the connections

claim that these two eigenvectors will

be in the column space or in other words

what we are claiming is that these three

columns can simply be written as a

linear combination of these two columns

and we are also sure that when we do a

times new one this will go to zero so

let us verify all of this in the next

slide

so let's first check eight times new one

so this is a matrix I have eight times

new one here and you can quite easily

see that when you do this computation

we'll get this zero zero zero which

basically shows that this is the eigen

vector corresponding to zero eigen value

interestingly in our initial lectures we

talked about null space and then we said

the null space vector identifies a

relationship between variables now since

this eigenvector is in the null space

the eigen vector corresponding to the

eigen vector corresponding to zero eigen

value or eigen vectors corresponding to

zero eigen values identify the

relationships between the variables

because these eigen vectors are in the

null space of the matrix so it's an

interesting connection that we can make

so the eigenvectors corresponding to

zero eigenvalue can be used to identify

relationships among variables now let's

do the last thing that we discuss let us

now check if the other two eigen vectors

shown below so this is for the other two

eigen values span the column space so

what I've done here is I've taken each

one of these columns from matrix a so

this is column one so this is a 1 this

is a 2 and this is a 3 column 1 is 6

times nu 2 column 2 is 8 times nu 2 and

column 3 is 2 times nu 3 so we can say

that a 1 a 1 a 2 and a 3

linear combinations of new to and new T

so new to a new tree form a basis for

this column space of matrix a so to

summarize we have ax equal to lambda X

and we largely focused on symmetric

matrices in this lecture so we saw that

if we have symmetric matrices they have

real eigenvalues we also saw that

symmetric matrices have n linearly

independent eigenvectors we saw the

daigon vectors corresponding to zero

eigenvalues span the null space of the

matrix a and eigenvectors corresponding

to nonzero eigenvalues span the column

space of a for symmetric matrices that

we described in this lecture so with

this we have described most of the

important fundamental ideas from linear

algebra that we will use quite a bit in

the material that follows the linear

algebra parts will be used in regression

analysis which you will see as part of

this course and many of these ideas are

also useful in algorithms that do

classification for example we talked

about how spaces and so on and the

notion of eigenvalues and eigenvectors

are used pretty much in almost every

data science algorithm of particular

node is one algorithm which is called

the principal component analysis where

these ideas of connections between null

space column space and so on are used

quite heavily so I hope that we have

given you reasonable understanding of

some of the important concepts that you

need to learn to understand some of the

material that we are going to teach in

this course and as I mentioned before

linear algebra is a vast topic there are

several ideas how do these ideas

translate which ones of these are

applicable are not applicable for non

symmetric matrices and so on and from

the previous lectures how do we develop

some of those concepts more can be found

in many good linear algebra books

however our aim here has been to really

call out the most important concepts

that we are going to use again and again

in this first course on data science for

engineers more advanced topics in linear

algebra will be covered when we teach

the next course on machine learning

where those concepts might be more

useful in advanced machine learning

techniques that we will pitch so with

this we close the series of lectures on

linear algebra and the next set of

lectures would be on the use of

statistics in data science I thank you

and I hope to see you back after you go

through the module on statistics which

will be taught by my colleague

precession connor seaman thank you

[Music]

[Music]