Vectors | Chapter 1, Essence of linear algebra

Summary

The video script introduces vectors as the foundational concept in linear algebra, exploring their interpretations from physics, computer science, and mathematics perspectives. It emphasizes the geometric view of vectors as arrows in a coordinate system, rooted at the origin, and their numerical representation as ordered lists or tuples of numbers. The script explains vector addition, a key operation in linear algebra, through the tip-to-tail method and its numerical equivalent, as well as scalar multiplication, which involves stretching or compressing vectors. The summary highlights the importance of understanding these operations for applications in data analysis, physics, and computer graphics, and sets the stage for further exploration of vector concepts like span, bases, and linear dependence.

Takeaways

• 📐 **Vectors Defined**: The script introduces vectors as fundamental to linear algebra, with three perspectives: physics, computer science, and mathematics.
• 🏋️‍♂️ **Physics Perspective**: In physics, vectors are arrows in space, defined by their magnitude and direction, with position flexibility.
• 💻 **Computer Science Perspective**: From a computer science viewpoint, vectors are ordered lists of numbers, where the order is significant.
• 🔢 **Mathematician's Perspective**: Mathematicians generalize vectors as entities that support vector addition and scalar multiplication, with an abstract approach.
• 📊 **Vector Addition**: Vectors are added by aligning their tails and drawing a new vector from the first tail to the second tip, representing the combined movement.
• 🔄 **Scalar Multiplication**: Multiplying a vector by a scalar (number) stretches or compresses it, with negative scalars reversing direction.
• 📈 **Geometric Interpretation**: The script emphasizes visualizing vectors as arrows in a coordinate system, rooted at the origin for clarity.
• 📝 **Coordinate System**: Coordinates are pairs or triplets of numbers that provide instructions for moving from the origin to the vector's tip in 2D or 3D space.
• 🔗 **Connection Between Views**: Linear algebra's power lies in the ability to switch between the geometric and numerical representations of vectors.
• 🌐 **Applications**: The script highlights linear algebra's utility in data analysis, physics, and computer graphics for pattern recognition and spatial description.

Q & A

What are the three perspectives on vectors mentioned in the script?

The script mentions three perspectives on vectors: the physics student perspective, the computer science student perspective, and the mathematician's perspective.

How does a physics student typically view vectors?

From a physics student's perspective, vectors are arrows pointing in space, defined by their length and direction, and can be moved around without changing their identity.

What is the computer science perspective on vectors?

In computer science, vectors are viewed as ordered lists of numbers, where the order of the numbers is significant.

How does a mathematician generalize the concept of vectors?

Mathematicians generalize vectors as anything that allows for the operations of vector addition and multiplication by a number, which are abstracted away from their specific representations.

Why are vector addition and multiplication by numbers important in linear algebra?

Vector addition and multiplication by numbers are fundamental operations in linear algebra because they form the basis for understanding more complex linear algebra concepts.

How is a vector's position in a coordinate system described?

A vector's position in a coordinate system is described by a pair of numbers in two dimensions or a triplet in three dimensions, which give instructions on how to move from the origin to the vector's tip.

What is the geometric interpretation of vector addition?

The geometric interpretation of vector addition involves placing the tail of the second vector at the tip of the first and then drawing a new vector from the tail of the first to the tip of the second, representing their sum.

What is the numerical representation of vector addition?

In numerical terms, vector addition involves matching corresponding components of the vectors and adding them together.

What does it mean to multiply a vector by a number?

Multiplying a vector by a number, or scalar multiplication, involves stretching or compressing the vector by that factor, or reversing its direction if the number is negative.

Why is the ability to switch between different perspectives of vectors important?

The ability to switch between different perspectives of vectors is important because it allows for the translation of concepts between the geometric and numerical domains, which is crucial for applications in data analysis, physics, and computer graphics.

What are some upcoming concepts in linear algebra that will be discussed after vectors?

Some upcoming concepts in linear algebra include span, bases, and linear dependence, which will be explored in subsequent videos.

Outlines

00:00

📐 Introduction to Vectors in Linear Algebra

This paragraph introduces vectors as the fundamental building block of linear algebra, explaining three perspectives on what vectors represent. From a physics student's viewpoint, vectors are arrows in space defined by their length and direction. In computer science, vectors are ordered lists of numbers, exemplified by a house pricing model where each house is a pair of numbers representing square footage and price. Mathematicians generalize these views, defining vectors as entities that can be added together and multiplied by numbers, operations that will be further discussed in the video. The paragraph emphasizes the importance of vector addition and multiplication by numbers in linear algebra and suggests thinking of vectors as arrows within a coordinate system, typically rooted at the origin. The concept of coordinates for vectors in two and three dimensions is also introduced, explaining how each pair or triplet of numbers corresponds to a unique vector.

05:00

🔢 Vector Operations: Addition and Scalar Multiplication

The second paragraph delves into the operations of vector addition and scalar multiplication. Vector addition is defined by the 'tip-to-tail' method, where the sum of two vectors is found by placing the tail of the second vector at the tip of the first and drawing a new vector from the tail of the first to the tip of the second. This operation is visualized as representing movements in space, analogous to adding numbers on a number line. The numerical representation of vector addition is also discussed, where coordinates are added component-wise. Scalar multiplication, or scaling, is explained through examples, showing how multiplying a vector by a number stretches, squishes, or reverses the vector. The paragraph concludes by emphasizing the importance of these operations in linear algebra and the utility of being able to translate between the geometric and numerical representations of vectors. The practical applications of these concepts in data analysis, physics, and computer graphics are also highlighted.

Keywords

💡Vector

A vector is a fundamental concept in linear algebra, representing both a direction and a magnitude in space. In the video, vectors are introduced from three perspectives: as arrows in space from a physics student's view, as ordered lists of numbers in a computer science context, and as abstract entities that can be added and scaled in a mathematician's framework. The video emphasizes the vector's role in linear algebra, highlighting its importance in operations such as addition and multiplication by a scalar.

💡Dimension

Dimension refers to the number of coordinates needed to specify a point in a space. In the script, two-dimensional vectors are mentioned as living in a flat plane, while three-dimensional vectors exist in the space we inhabit. The concept is crucial for understanding the structure and properties of vectors and their operations within different spaces.

💡Coordinate System

A coordinate system is a grid of intersecting lines or planes used to specify the position of points in a space. The video describes the xy-plane with an origin at the intersection of the x-axis and y-axis, which serves as the starting point for vectors. Understanding the coordinate system is essential for visualizing and calculating vector operations.

💡Origin

The origin is the point of intersection in a coordinate system, typically considered as the starting point for vectors. In the video, it is mentioned that in linear algebra, vectors are often rooted at the origin, which contrasts with the physics perspective where vectors can be positioned anywhere in space.

💡Vector Addition

Vector addition is the operation of combining two vectors to produce a third vector. The video explains this through the 'tip-to-tail' method, where one vector is positioned so that its tail is at the tip of the other, and a new vector is drawn from the tail of the first to the tip of the second. This operation is central to understanding how vectors represent movements or changes in space.

💡Scalar Multiplication

Scalar multiplication involves multiplying a vector by a number, or scalar, to stretch or compress it. The video illustrates this by examples such as doubling the length of a vector or reducing it to one-third. This operation is key to understanding how vectors can be scaled, which is a fundamental concept in linear algebra.

💡Components

Components of a vector refer to the individual elements of its ordered list representation. In the video, the components of a vector are the numbers that specify its position in a coordinate system. The script uses the example of a two-dimensional vector with coordinates (1, 2) to explain how vector operations are performed numerically.

💡Scalar

A scalar is a number that can be used to scale a vector, changing its length without affecting its direction. The video explains that scalars are used in operations like scalar multiplication, where they can stretch, compress, or reverse the direction of a vector. Scalars are essential in linear algebra for their role in vector transformations.

💡Abstraction

Abstraction in the context of the video refers to the mathematician's approach to vectors, where they are considered as entities that can be added and scaled, independent of their representation. The video suggests that this abstract view is important for understanding the general principles of linear algebra, which can then be applied to various representations of vectors.

💡Linear Algebra

Linear algebra is a branch of mathematics that deals with linear equations, linear transformations, and vector spaces. The video emphasizes that linear algebra revolves around operations like vector addition and scalar multiplication, which are fundamental to understanding the subject. The script also highlights the practical applications of linear algebra in fields like data analysis, physics, and computer graphics.

💡Span

Although not explicitly defined in the script, the concept of span is mentioned as a topic to be covered in future videos. Span refers to the set of all possible linear combinations of a given set of vectors, which is a fundamental concept in understanding the structure of vector spaces. The video hints that span will be important for understanding how vectors can be combined to generate new vectors within a space.

Highlights

Vectors are fundamental to linear algebra, with three distinct perspectives: physics, computer science, and mathematics.

In physics, vectors are arrows in space defined by length and direction.

Computer science views vectors as ordered lists of numbers, crucial for analytics and modeling.

Mathematicians generalize vectors for operations like addition and scalar multiplication.

Vectors in linear algebra are often rooted at the origin, differing from the physics perspective.

Coordinate systems, like the xy-plane, are essential for understanding vector operations.

The coordinates of a vector provide instructions from the origin to its tip.

Vectors are distinguished from points by writing their coordinates vertically with square brackets.

In three dimensions, vectors are associated with ordered triplets of numbers.

Vector addition is defined by the tip-to-tail method, representing combined movements.

Vector addition numerically involves matching terms and adding them together.

Scalar multiplication stretches, squishes, or reverses the direction of a vector, called scaling.

In numerical terms, scalar multiplication corresponds to multiplying each vector component by the scalar.

Linear algebra's usefulness lies in the ability to translate between geometric and numerical vector representations.

Linear algebra provides a visual way to conceptualize data and describe space for various applications.

Upcoming videos will explore concepts like span, bases, and linear dependence in the context of vectors.

Transcripts

The fundamental, root-of-it-all building block for linear algebra is the vector.

So it's worth making sure that we're all on the same page about what exactly a vector is.

You see, broadly speaking, there are three distinct but related ideas about vectors,

which I'll call the physics student perspective,

the computer science student perspective, and the mathematician's perspective.

The physics student perspective is that vectors are arrows pointing in space.

What defines a given vector is its length and the direction it's pointing,

but as long as those two facts are the same, you can move it all around,

and it's still the same vector.

Vectors that live in the flat plane are two-dimensional,

and those sitting in broader space that you and I live in are three-dimensional.

The computer science perspective is that vectors are ordered lists of numbers.

For example, let's say you were doing some analytics about house prices,

and the only features you cared about were square footage and price.

You might model each house with a pair of numbers,

the first indicating square footage and the second indicating price.

Notice the order matters here.

In the lingo, you'd be modeling houses as two-dimensional vectors,

where in this context, vector is pretty much just a fancy word for list,

and what makes it two-dimensional is the fact that the length of that list is two.

The mathematician, on the other hand, seeks to generalize both these views,

basically saying that a vector can be anything where there's a sensible notion of adding

two vectors and multiplying a vector by a number,

operations that I'll talk about later on in this video.

The details of this view are rather abstract, and I actually think it's healthy to ignore

it until the last video of this series, favoring a more concrete setting in the interim.

But the reason I bring it up here is that it hints at the

fact that the ideas of vector addition and multiplication by

numbers will play an important role throughout linear algebra.

But before I talk about those operations, let's just settle in

on a specific thought to have in mind when I say the word vector.

Given the geometric focus that I'm shooting for here,

whenever I introduce a new topic involving vectors,

I want you to first think about an arrow, and specifically,

think about that arrow inside a coordinate system, like the xy-plane,

with its tail sitting at the origin.

This is a little bit different from the physics student perspective,

where vectors can freely sit anywhere they want in space.

In linear algebra, it's almost always the case

that your vector will be rooted at the origin.

Then, once you understand a new concept in the context of arrows in space,

we'll translate it over to the list of numbers point of view,

which we can do by considering the coordinates of the vector.

Now, while I'm sure that many of you are already familiar with this coordinate system,

it's worth walking through explicitly, since this is where all of the important

back and forth happens between the two perspectives of linear algebra.

Focusing our attention on two dimensions for the moment,

you have a horizontal line, called the x-axis, and a vertical line, called the y-axis.

The place where they intersect is called the origin,

which you should think of as the center of space and the root of all vectors.

After choosing an arbitrary length to represent one,

you make tick marks on each axis to represent this distance.

When I want to convey the idea of 2D space as a whole,

which you'll see comes up a bit in the way, but right now they'll get a

little bit in the way.

The coordinates of a vector is a pair of numbers that basically gives

instructions for how to get from the tail of that vector at the origin to its tip.

The first number tells you how far to walk along the x-axis,

positive numbers indicating rightward motion, negative numbers indicating leftward

motion, and the second number tells you how far to walk parallel to the y-axis

after that, positive numbers indicating upward motion,

and negative numbers indicating downward motion.

To distinguish vectors from points, the convention is to write

this pair of numbers vertically with square brackets around them.

Every pair of numbers gives you one and only one vector,

and every vector is associated with one and only one pair of numbers.

What about in three dimensions?

Well, you add a third axis, called the z-axis,

which is perpendicular to both the x and y-axes, and in this case,

each vector is associated with ordered triplet of numbers.

The first tells you how far to move along the x-axis,

the second tells you how far to move parallel to the y-axis,

and the third one tells you how far to then move parallel to this new z-axis.

Every triplet of numbers gives you one unique vector in space,

and every vector in space gives you exactly one triplet of numbers.

All right, so back to vector addition and multiplication by numbers.

After all, every topic in linear algebra is going to center around these two operations.

Luckily, each one's pretty straightforward to define.

Let's say we have two vectors, one pointing up and a little to the right,

and the other one pointing right and down a bit.

To add these two vectors, move the second one so

that its tail sits at the tip of the first one.

Then, if you draw a new vector from the tail of the first one to

where the tip of the second one sits, that new vector is their sum.

This definition of addition, by the way, is pretty much the only time

in linear algebra where we let vectors stray away from the origin.

Now, why is this a reasonable thing to do?

Why this definition of addition and not some other one?

Well, the way I like to think about it is that each vector represents a certain movement,

a step with a certain distance and direction in space.

If you take a step along the first vector, then take a step in the direction

and distance described by the second vector, the overall effect is just

the same as if you moved along the sum of those two vectors to start with.

You could think about this as an extension of

how we think about adding numbers on a number line.

One way that we teach kids to think about this, say with 2 plus 5,

is to think of moving two steps to the right followed by another five steps to the right.

The overall effect is the same as if you just took seven steps to the right.

In fact, let's see how vector addition looks numerically.

The first vector here has coordinates 1, 2, and the second one has coordinates 3,

negative 1.

When you take the vector sum using this tip-to-tail method,

you can think of a four-step path from the origin to the tip of the second vector.

Walk 1 to the right, then 2 up, then 3 to the right, then 1 down.

Reorganizing these steps so that you first do all of the rightward motion,

then do all the vertical motion, you can read it as saying first

move 1 plus 3 to the right, then move 2 minus 1 up.

So the new vector has coordinates 1 plus 3 and 2 plus negative 1.

In general, vector addition in this list of numbers conception

looks like matching up their terms and adding each one together.

The other fundamental vector operation is multiplication by a number.

Now this is best understood just by looking at a few examples.

If you take the number 2 and multiply it by a given vector,

it means you stretch out that vector so that it's two times as long as when you started.

If you multiply that vector by, say, one-third,

it means you squish it down so that it's one-third the original length.

When you multiply it by a negative number, like negative 1.8,

then the vector first gets flipped around, then stretched out by that factor of 1.8.

This process of stretching or squishing or sometimes reversing the direction of

a vector is called scaling, and whenever you catch a number like two or one-third

or negative 1.8 acting like this, scaling some vector, you call it a scalar.

In fact, throughout linear algebra, one of the main things that numbers do is scale

vectors, so it's common to use the word scalar pretty much interchangeably with the word

number.

Numerically, stretching out a vector by a factor of, say, 2,

corresponds with multiplying each of its components by that factor, 2.

So in the conception of vectors as lists of numbers,

multiplying a given vector by a scalar means multiplying each one of those

components by that scalar.

You'll see in the following videos what I mean when I say linear algebra topics tend to

revolve around these two fundamental operations,

vector addition and scalar multiplication.

And I'll talk more in the last video about how and why the

mathematician thinks only about these operations,

independent and abstracted away from however you choose to represent vectors.

In truth, it doesn't matter whether you think about vectors as fundamentally being arrows

in space, like I'm suggesting you do, that happen to have a nice numerical

representation, or fundamentally as lists of numbers that happen to have a nice geometric

interpretation.

The usefulness of linear algebra has less to do with either one of these

views than it does with the ability to translate back and forth between them.

It gives the data analyst a nice way to conceptualize many lists

of numbers in a visual way, which can seriously clarify patterns

in data and give a global view of what certain operations do.

And on the flip side, it gives people like physicists and computer

graphics programmers a language to describe space and the computer.

When I do math-y animations, for example, I start by thinking about what's

actually going on in space, and then get the computer to represent things numerically,

thereby figuring out where to place the pixels on the screen.

And doing that usually relies on a lot of linear algebra understanding.

So there are your vector basics, and in the next video I'll start getting into some

pretty neat concepts surrounding vectors, like span, bases, and linear dependence.

See you then!