Main content

## Algebra (all content)

### Course: Algebra (all content) > Unit 20

Lesson 10: Matrices as transformations- Transforming vectors using matrices
- Use matrices to transform 3D and 4D vectors
- Transforming polygons using matrices
- Transform polygons using matrices
- Matrices as transformations
- Matrix from visual representation of transformation
- Visual representation of transformation from matrix
- Understand matrices as transformations of the plane

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Transforming vectors using matrices

Sal transforms a 2-dimensional vector using a 2x2 matrix, and draws the original vector and its image on the plane. Created by Sal Khan.

## Want to join the conversation?

- What's the meaning of a transformation matrix? What does it represent?(14 votes)
- At4:16what did Sal mean by saying 'P Prime'?

I'm a bit confused at that part, could someone please explain it to me?

Thanks!(4 votes)- P is the original vector.

P' is the transformed vector.

He could have called the transformed version of vector P P1 or Pt, just as to differentiate it from the original vector. Often the ' symbol is used to denote a transformation of a variable by some process.

Very soon you will learn a very different meaning for the symbol ' within the context of differential calculus.(1 vote)

- Can there be variables in matrices?

And if so, Can there be matrix equations?

Ex:`[ 3 5 ] [ 3 5 ]`

[ 4 x ] = [ 9 -5 ]

[ 3 3 ] [ 1/2 5 ]**Note: The separate square brackets are actually for 3 X 3 matrices**.(3 votes)- Yes! We use matrices to solve equations - and you will get there very soon.

Here is a sneak peak:

https://www.khanacademy.org/math/precalculus/precalc-matrices/matrix-equations/v/matrix-equations-systems(5 votes)

- What do the values in the transformation matrix mean? i.e in case of the vector's column matrix, 2 is the x component whereas 1 is the y component, so in a similar fashion what do the values in transformation matrix signify? How do they tell that x-component should translate to 5 and y component to 0?1:25(4 votes)
- How can we actually determine the Transformation matrix?(3 votes)
- We get to pick whatever transformation matrix we want depending on our particular goal. See the remaining videos of this section.(3 votes)

- In the P column matrix of order 2 cross 1, in the above video, how do we figure out which number is x and which one is y?(2 votes)
- it is conventional that first is X and next one is Y. Thus P_{1,1 } is X and P_{2,1} is Y(4 votes)

- Is the transformation matrix always the first term ?(3 votes)
- Is there a specific reason you had to multiply the vector with the matrix? Adding a matrix, subtracting a matrix, and using operations within the vector matrix affect the vector too.(2 votes)
- I did not see how 3x3 matrices determinate or how to find the inverse of anything greater than a 2x2. Where can if find more information on transformation matrices ie. mirroring, rotation etc(2 votes)
- what do i do if id dont know what on earth a vector is(2 votes)
- You go to the lessons for Vectors, then do the section on Matrices.

Here is vectors: https://www.khanacademy.org/math/precalculus/vectors-precalc(1 vote)

## Video transcript

Voiceover:Let's say that
we've got a position vector, P and it is equal to or
represented as a column vector, right over here, 2, 1. If we wanted to plot this,
and that is what I'll do. Let's plot it. If we were to plot it, that's my Y axis, this right over here is my X axis. If we assume that the first entry here is X coordinate, 1, 2, 1. Our position vector is
going to be right over here. We could represent it. We could represent it in this kind of vector symbol, like that. Where the tail is at the origin and the tip, the head, is at that point or we could say it's really
representing this position, right over here on the coordinate plan. What I wanted to do in this video is apply a transformation
to this position vector. The way I'm going to do it is I'm going to multiply our vector, P by a matrix and then the resulting product is going to give me
another position vector. So, what do I mean by that? I could have a transformation
matrix, capital T and let's just say it is
equal to 2, 1, -1, and 2. What happens if I multiply TxP? Let's do that right over here. TxP, so first let's just make sure that this is actually a valid operation that matrix multiplication or matrix vector
multiplication is defined here. Let's just look at what T and P look like. I'm gonna copy and paste. That is T and this is P right over here. Can we multiply a 2x2 matrix times a column vector like this, which is essentially 2x1 matrix. Well sure, as as we know matrix multiplication is only defined, or at least conventional
matrix multiplication is only defined if the first matrix number of columns is equal to the number of rows in the second
matrix, right over here. We see there, both of those are 2. This is going to result in a 2x1 matrix. This is going to result in a 2x1 matrix. What's interesting about that is that this is another column vector. This is another position vector. You took this vector P, multiplied it by this
transformation vector and we're gonna get another 2x1, which we could think of
as a position vector, which we could plot. What's essentially happened is this transformation vector has moved, has essentially given this point. It has given us a new point. Let's think about what that is. This first entry here, we're essentially thinking about, we're going to deal with the first row and the first and the only column, really. Let me do, use a color
that I haven't used yet. We're gonna think about
this row and this column. It's gonna be 2x2, which is 4, plus 1x1, which is 1 or this is just going to be equal to 5. For the second entry, right over here. It's going for the
second row, first column. It's gonna deal with the second row here and the one and only
column, right over here. -1x2 is -2, plus 2x1, which is 2. -2+2 is just equal to 0. Now, we have the position 5, 0, which is right over here. 1,2,3,4,5 So, we started with this point, this position vector, I
guess you could say, P As transformed it to this
other position vector and I guess we could call this P prime. If you wanted to draw them as vectors, this right over, kinda
tradition vector form, that right over there is P prime and this right over here is P. This right over here is P. This is P actually, let me
make sure there's a prime. This is P prime and the way we got from P to P prime is using this transformation matrix.