Lecture 05 : Basis and Dimension
Summary
TLDRThis lecture delves into the fundamental concepts of basis and dimension in vector spaces, crucial for machine learning algorithms and applications like dimension reduction and dictionary learning. It explains the criteria for a set of vectors to form a basis, emphasizing linear independence and the ability to span the space. The lecture also distinguishes between finite and infinite dimensional spaces, provides examples of bases in various spaces, and introduces the concept of dimension as the number of vectors in a basis. Important results related to basis and dimension, including the implications for linear dependence and the extension of linearly independent sets, are highlighted.
Takeaways
- 📚 The lecture discusses the concept of basis and dimension in vector spaces, which are fundamental to machine learning algorithms.
- 🔍 A basis for a vector space is a set of linearly independent vectors that span the entire space, allowing any vector to be expressed as a linear combination of these basis vectors.
- 📏 There are two types of vector spaces: finite-dimensional and infinite-dimensional, differentiated by the number of vectors in their basis.
- 🧩 The dimension of a vector space is the number of vectors in its basis, which is a key characteristic of the space.
- 📉 In machine learning, basis and dimension concepts are crucial for algorithms such as dimension reduction and dictionary learning.
- 🌐 The standard basis for R2, R3, and RN are sets of vectors with elements that are zero except for a single one, representing the standard orientation in space.
- 🔢 The dimension of spaces like R2, RN, and the space of all 2x2 real matrices are given by simple formulas related to the number of elements in the basis.
- 🔑 If a set of vectors contains more than the dimension of the vector space, it is linearly dependent, emphasizing the importance of the basis size.
- 📈 Any linearly independent subset of a vector space can be extended to form a basis for that space, highlighting the flexibility in choosing a basis.
- 📊 For finite-dimensional vector spaces, all bases contain the same number of vectors, which equals the space's dimension.
- 📘 The lecture provides examples of finding the basis and dimension of subspaces in R3 and R4, illustrating the process of determining these properties.
Q & A
What is the main topic of the lecture?
-The main topic of the lecture is the concept of basis and dimension in vector spaces, which are important in the context of machine learning algorithms.
Why are basis and dimension important in machine learning?
-Basis and dimension are important in machine learning because they help in representing input data as a linear combination of certain vectors, which is essential for algorithms like dimension reduction and dictionary learning.
What is the definition of a basis in vector spaces?
-A basis for a vector space V is a set of linearly independent vectors in V that spans the vector space V. This means every vector in the space can be written as a linear combination of the basis vectors.
What are the two main properties a set of vectors must have to be considered a basis?
-A set of vectors must be linearly independent and must span the entire vector space to be considered a basis.
What is the difference between finite and infinite dimensional vector spaces?
-A finite dimensional vector space has a basis containing a finite number of vectors, while an infinite dimensional vector space has a basis with an infinite number of vectors.
Can you provide an example of a basis for R2 over the real numbers?
-An example of a basis for R2 over the real numbers is the set of vectors {(1, 0), (0, 1)}.
What is the dimension of a vector space?
-The dimension of a vector space is the number of elements in a basis of the vector space.
What is the relationship between the dimension of a vector space and the number of vectors in its basis?
-The dimension of a vector space is equal to the number of vectors in any of its bases.
How can you determine if a set of vectors is linearly dependent?
-A set of vectors is linearly dependent if it contains more vectors than the dimension of the vector space it is supposed to span.
Can a linearly independent set of vectors be extended to form a basis of a vector space?
-Yes, if a linearly independent set of vectors does not span the entire vector space, it can be extended by adding more linearly independent vectors until it becomes a basis.
What is the significance of the dimension of the intersection of two subspaces in relation to the dimensions of the subspaces themselves?
-The dimension of the intersection of two subspaces, along with the dimensions of the subspaces themselves, follows the formula: dimension of S1 + dimension of S2 = dimension of (S1 + S2) + dimension of (S1 ∩ S2).
Outlines
📚 Introduction to Basis and Dimension in Vector Spaces
The script begins with a refresher on vector subspaces and introduces the concept of basis and dimension of these spaces, crucial for machine learning algorithms. It explains that feature vectors can be represented as linear combinations of certain vectors, which may be linearly independent or orthogonal. The lecture aims to identify the specific set of vectors that form a basis for the feature space, which is essential in algorithms like dimension reduction and dictionary learning. The mathematical definition of a basis is presented, emphasizing that it consists of linearly independent vectors that span the vector space. The script also distinguishes between finite and infinite dimensional vector spaces.
🔍 Examples of Vector Space Bases and Their Properties
This paragraph provides examples of vector space bases, such as the standard basis for R2, R3, and RN, which are sets of vectors with elements of one and zeros elsewhere. It further explores the basis of spaces of 2x2 matrices with real entries and 2x2 symmetric matrices, highlighting the difference in the number of basis vectors required for each. The dimension of a vector space is defined as the number of elements in its basis, with examples given for R2, RN, and spaces of matrices and polynomials, emphasizing that the dimension is a key characteristic of the space.
📉 Important Results on Basis and Dimension
The script discusses important results related to bases and dimensions of vector spaces. It states that if a set of vectors exceeds the dimension of the space, it must be linearly dependent. It also mentions that any linearly independent subset can be extended to form a basis of the vector space. Additionally, it asserts that all bases of a finite-dimensional vector space contain the same number of vectors, which equals the space's dimension, and introduces the concept of the sum and intersection of subspaces and their dimensions.
📚 Finding the Basis and Dimension of Subspaces
The paragraph demonstrates how to find the basis and dimension of subspaces using the example of a subspace S of R3, defined by a linear equation. It shows the process of simplifying the equation to express the components of vectors in S in terms of a smaller set of variables, leading to the identification of the basis vectors and the calculation of the subspace's dimension. The example illustrates the method of determining the basis and dimension for subspaces within a larger vector space.
🔍 Further Examples of Subspace Analysis
This section continues with more examples of finding the basis and dimension of subspaces within R3 and R4. It explains how to determine the basis for a subspace defined by a set of equal components and how to find the intersection of two subspaces. The script uses these examples to show that the dimension of the intersection of two subspaces can be zero, indicating that their only common element is the zero vector, and that the sum of dimensions of two subspaces equals the dimension of their union plus the dimension of their intersection.
🚀 Conclusion and Preview of Upcoming Topics
The script concludes with a summary of the concepts of basis and dimension learned in the lecture and previews the next topic, which will be linear transformations, another fundamental concept in mathematics and machine learning. It also provides references for further study and expresses gratitude for the audience's attention, ending with applause and music, signaling the end of the lecture.
Mindmap
Keywords
💡Vector Subspaces
💡Basis
💡Dimension
💡Linearly Independent
💡Linear Combination
💡Finite Dimensional Vector Spaces
💡Infinite Dimensional Vector Spaces
💡Standard Basis
💡Orthonormal
💡Linear Dependence
💡Subspace
Highlights
Introduction to the concept of basis and dimension in vector subspaces, crucial for machine learning algorithms.
Basis vectors must be linearly independent and span the vector space.
Explanation of finite and infinite dimensional vector spaces.
The importance of basis in dimension reduction and dictionary learning algorithms.
Wavelet-based algorithms use orthonormal functions as a basis to approximate functions.
Mathematical definition of a basis for a vector space V over a field F.
Qualifications for a set to be a basis: linear independence and spanning the vector space.
Examples of bases for R2, R3, and RN with standard basis vectors.
Basis for 2x2 matrices and symmetric matrices as vector spaces.
Dimension of a vector space is the number of elements in its basis.
Dimension examples for R2, RN, and spaces of 2x2 matrices and polynomials.
Result: A basis of an n-dimensional vector space cannot have more than n vectors.
Result: Any linearly independent subset can be extended to form a basis.
Result: All bases of a finite-dimensional vector space have the same number of vectors.
Result: Dimension of the sum of two subspaces equals the sum of their dimensions minus the dimension of their intersection.
Finding the basis and dimension of a subspace defined by a linear equation in R3.
Example of finding the basis and dimension for subspaces S1 and S2 in R4.
Conclusion on the importance of understanding basis and dimension for further study in linear transformations.
Transcripts
[Music]
[Music]
if you remember in the last lecture we
have talked about uh Vector sub
spaces so today we are again going to
talk about a very important concept from
the vector spaces that is basis and
dimension of vector sub
spaces so in most of the machine
learning algorithm we need to
represent our input data that is in
terms of feature vectors as a linear
combination of certain
vectors that all the feature vectors we
want to write as linear combination of a
set of vectors those set of vectors may
be linearly independent or they may be
orthogonal so to write all the feature
vectors of our data set in terms of
linear combination of these vectors we
have to find out that particular set of
vectors so these vectors form a basis
for the feature space today we will
learn about the basis of the vector
spaces the concept of basis is really
important especially in the case of
Dimension
reduction or the recent algorithm like
dictionary learning based algorithm
further in wavelet based algorithms we
approximate any function as the linear
combination of set of orthonormal
functions those we have generated from
the
mother wavelength so let us come to the
definition of basis so
mathematically it's speaking the
definition of bases is given as let VF
be a vector space so we are having a
vector space V over the field F A basis
for V is a set of linearly independent
vectors in V which spans the vector
space V now so what
there should what should be the
qualification to be a basis the first
all the vectors of that set should be
linearly
independent number
two you take any Vector from the vector
space V that Vector can be written as
the linear combination of the vectors of
the basis set there are two types of
vector spaces finite dimensional Vector
spaces and infinite dimensional Vector
spaces so if the basis of a vector space
V contains the finite number of vectors
then we say that it is a finite
dimensional Vector space otherwise we
say that the vector space is an infinite
dimensional Vector
space a vector in basis is called the
basis Vector so if we talk this
definition
mathematically so what we are having a
set
B having vectors let us say V1
V2 up to
VN is a
basis
of a n dimensional Vector space
we if the set of
vectors V1 V2
VN is linearly
independent
that is the first thing we
need and the second thing
is for any
vector v belongs to the vector space
V we
have V =
to Alpha 1 V1 + Alpha 2
V2 plus alpha n
VN
where alpha
1 Alpha
2 alpha
n
are
scalars from the field
f
so this is uh mathematically we can
Define basis in this way also so if you
see some example of the
bases
so if you take V equals
to R2 over the field r
then if you take vectors like 1
Z and 01 in
R2 then this set forms a basis 4 V the
first thing both of these vectors are
linearly
independent and the second thing you
take any vector v from R2 we can write
that Vector let us say you are taking
some vector Alpha Beta arbitrary
Vector belongs to
R2
then we can write Alpha Beta as Alpha
* 1 + beta *
01 so what I want to say that this set
spans whole R2 space similarly if you
take V = to
R3 over the field of real
numbers then one of the possible basis
is 1 0 0 0 1 0 and 0 0
1 if you take V equals to
RNR then one of the possible basis is 1
0 0
0 0
1
0 0
0
1 so all these are nles vector having
one of the element is one and rest of
the elements are zero so all these three
are called standard basis for R2 R3 and
RN respectively if we
talk if we take a vector space like
this so R 2 by two matrices having real
entri
so
so certainly this V over the field of
real number forms a vector
space we have seen it in previous
lectures now what will be the basis of
this so basis of this will
be
so this is one of the possible basis for
this Vector space V here you can notice
all the vectors those are 2x2 matrices
are linearly
independent and any 2x2 matrix can be
written as the linear combination of
these four matrices
if I change this uh Vector space let us
say V equals
to m2x
2 set of
all
2x2
real symmetric
Matrix then what will be the basis
certainly this set over the field of
real numbers will form a vector space
and what will be the basis one of the
possible bases will be given
as 1 0
1 sorry 1 0 0
0 0 1 1
0 and 0 0 0
1 because this is space of symmetric
matrices so these two elements will be
equal to have symmetric to be the matric
symmetric so this basis will be having
three vectors while this is having four
vectors so these are some of the
examples of basis of different Vector
spaces my next definition is dimension
so what is the dimension of a vector
space
the dimension of a vector space is
nothing just the number of elements in
the
bases or number of vectors in the
basis so formally I can say the number
of vectors in a basis of VF is called
the dimension of the vector space VF so
as you have seen example uh the
dimension of vector space R2 over the
field of real number is two the
dimension of vector space RN over the
field of real number is n the dimension
of the vector space of all 2 by two real
matrices over the field of real number
is four if you are taking the vector
space of all M by n real matrices over
the field of real number then Dimension
will become M * n if you take the vector
space of all the pols having degree n or
less then the dimension of that Vector
space will be n + 1 because we will be
having n + 1 elements in the
basis similarly dimension of vector
space of all pols is
infinite why because it is a infinite
dimensional Vector
space we can write any Pol omal from
this Vector space as a linear
combination of finite number of vectors
from
that basis however basis will be
containing the infinite number of
elements now come to some important
results related to the basis and
dimension so my first result is let V1
V2 VN be a basis of a n dimensional
Vector space VF if s be a subset of
vectors having more than n
vectors then the set s is linearly
dependent so what I want to convey that
if the dimension of vector space is n
any set containing more than n vectors
will be linearly dependent because in a
set you can have at most and linearly
independent vector v
s and if you are having such a set where
you are having an linearly independent
Vector then that set will be a basis of
that particular Vector space so what in
other words I can say that the basis is
a maximal linearly independent set of a
vector space if you are having any other
vector in that set that Vector can be
written as a linear combination of rest
of the vectors my second result is let
V1 V2 VK be a linearly independent
subset of a n dimensional Vector space V
over the field F where K is less than
n then s can be extended to a basis of
VF for example suppose I am having a
Vector space R5 and I am having three
linearly independent vectors of R5 so
what I can do I can include two more
Alli vectors to that set of three
vectors and then I will be having five
linearly independent
vectors in
R5 and then this set will be a basis of
R5 another important result is like let
VF be a finite dimensional Vector space
then any two bases of V have the same
number of
vectors we can have multiple bases for a
vector space
however each of the bases will be having
the same number of
vectors that is equals to the dimension
of the vector space so for example if we
talk
about
r3r you have seen
that one of the set of vector 1 0
0 0 1
0 and 0 0
1 this particular set forms a basis for
R3 over the field of real numbers
if I take another set of three vectors
in R3 let us say 1 0
0 1 1
0 and 1 one
1 so again these are three vectors in
R3 and all this is a linearly
independent set
so B2 also a basis for R3 so we can have
many other basis where we are having
three Ali vectors from the set R3 from
the vector space R3 as a basis but the
common thing is all these sets will be
having three
vectors my next result is let V be a
vector space over the field F and it is
finite
dimensional if you take two sub species
of V let us say S1 and S2 then dimension
of S1 + dimension of S2 equal to
dimension of S1 + S2 Plus dimension of
S1 intersection S2 so we have written S1
here it will be S2 so dimension of S1
intersection S2 we will see this result
by this
example suppose we need to find out the
basis and dimension of the Subspace s of
R3 given Ed S = to X1 X2
X3 such that X1 + X2 - X3 = to 0 and we
are having another Subspace of R3 that
is W given by vectors X1 X2 X3 belongs
to R3 such that
X1 = to X2 = to X3 so how to find basis
of the so my Vector space
is R3 over the field of real
numbers and I am defining my set S
as the space of all the vectors X1 X2
X3 belongs to R3 such
that such that X1 +
X2 - X3 = to
0
so in the previous lecture we have
learned that how to prove that s is a
Subspace of
R3 now we need to find out basis of s so
here I can
write X1 X2
X3 belongs to
R3 such that X1 +
X2 =
to
X3 so I can write it
X1
X2 and since X3 = to X1 + X2 so I can
replace X3 as X1 +
X2
this I can write
as
X1
1 Z so one is coming from the first
component in second component we don't
have any X1 so X1 is zero and then
1
plus
X2 0 1
1 so here I can write the basis of
s is containing two vectors 1
1 and 0 1
1 hence dimension of
s equals to 2 so in that way we can find
out the dimension basis and dimension of
a vectory space take another example
there again we equals
to
R3 over the field of real number and we
are having w
h X1 X2
X3 such that X1 = to
X2 equals
to
X3 so what kind of Vector we are having
in W where all the three
components are equal so for example 1 1
1 2 2 2 -1 -1 -1 all these kind of
vectors so certainly the basis of I can
write it
X1
X1 X1 because X2 equals to X1 and X3 is
also
equal to
X1 belongs to
R3 so what is this it is
X1 1 1
1 so the basis of w
is having only one vector 1 one 1 so any
Vector of w can be written as some
scalar times this 1 one one so dimension
of w
is
1 if you need to find out basis
of s intersection W then what is s
intersection
W it is all vectors X1 X2
X3 belongs to
R3 such that at X1 +
X2 - X3 = 0 this condition is coming
from the Subspace
s
and the condition from Subspace W is X1
= to X2 = to
X3 so this I can
write X1
X2 X1 + X2 such that this I have written
by this condition such that X1 = to X2 =
to
X3 so if I take X1 = to
X2 then I can write it X1 now X2 = to X1
so
X1 and then X1 + X1
is
2x1 however what I need I need all the
three component should be
equal so in this case what we can have
the only possibility
is that it should have the zero
Vector because we need X1 X2 and X1 + X2
all three are equal it will be only
where then X1 is 0 X2 is0 and so that X1
+ X2 also become Z so this is the
basis of s intersection W hence
dimension
of s intersection W
is zero why zero because I told you
earlier
that the vector space containing the
zero Vector can be span by the empty set
by the basis Pi empty basis so there is
no element in the basis hence basis is
dimension of s intersection W is zero so
this is the way for finding the
basis and then dimension of a vector
space let us take one more example find
the basis and dimension of S1 S2 S1
intersection
S2 where S1 S2 are the subspaces of R4
over the field of real
numbers as I told you the intersection
of two subspaces also a Subspace so
again S1 intersection S2 is also a
Subspace of
R4 so let us find out the basis of S1 so
S1 is given as X X1
X2 X3
X4 belongs to R4 such that the first
constant is X1 + X2 -
X3 + X4 =
0 the second condition is X1 +
X2 + X3 + X4 =
to0 so what we are having here by adding
these two conditions what I can write X4
=
to - X1 -
X2 now from the first condition I can
write X3 = to X1 +
X2 plus
X4 if I substitute the value of X4 from
here that is - X1 - X2 I can get X3 = to
0 so let me write it here
Now
using these result so
X1 X2 X3 0 and X4 is - X1 -
X2 so it
is if I take X1 it is 1 0 0 - 1
+ X2 0 1 0 - 1
so here basis of S1
is 1 0 0
-1 and then 0 1 0 -
1 hence the dimension of S1 is 2
similarly you can find the dimension of
S2 using the same procedure and you will
find that dimension of S2 is coming 1 2
1 and 0 1 1 2
so hence basis of S2
is given by these two vectors and
dimension of S2 is 2 so now dimension of
S1 is 2 dimension of S2 is 2 so
dimension of S1 plus dimension of S2 is
4 which is equals to the vector space
dimension of the vector space R4 hence
dimension of S1 intersection S2 is zero
so the basis of S1 intersection S2
contains only zero
element which says that if you put all
these four conditions together the
solution will be X1 = to X2 = to X3 = to
X4 = to 0 all four components are zero
so in this lecture we have learned the
concept of basis and
dimension in the next lecture we will we
learn another very important concept of
mathematics related to the machine
learning that is linear
Transformations these are the
references for this
lecture hope you have enjoyed the
lecture thank you thank you very
[Applause]
[Music]
much
[Applause]
[Music]
[Laughter]
n
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