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### Course: Linear algebra>Unit 2

Lesson 3: Transformations and matrix multiplication

# Compositions of linear transformations 2

Providing the motivation for definition of matrix products. Created by Sal Khan.

## Want to join the conversation?

• At , Sal says that a1 belongs to Rn, shouldn't it be Rm
• Yes, in the example given , m represents the number of rows and rows travel from top to bottom.
• I don't get it? Is every column in a matrice really a vector in disguise?
• Yes. The converse is true as well, every vector is a matrix in disguise, e.g. a column vector with 5 entries is really a 5 x 1 matrix. :D
Thumbs up!
• How come can we take the same vector x both for S and T transformations? Especially when the corresponding co-domains are of different dimensions? I mean the statements T(x)=Bx and S(x)=Ax. Or am I just saying bollocks?
• x is just the variable used for the functions. x isn't going to be the same for both T and S, they just both use x as the placeholder for the input. You'll notice when he does T(S( x )) that the x inside the T function was replaced with S( x ), but that doesn't mean that x = S( x ), it just means that the output of S is being used as the input for T.
• How do you prove that the combination of the composition of two given linear transformations is also a linear transformation? Let's say, V---->W , W------>U , is V------->U also a linear transformation?
(1 vote)
• Let's call V->W A, W->U B and V->U C.

Since we know A and B are linear transformations, we know that
A(x + y) = A(x) + A(y) and A(cx) = cA(x)
and similarly for B
B(x + y) = B(x) + B(y) and B(cx) = cB(x)

And now we want to prove that C(x) = B(A(x)) is a linear transformation. The same conditions apply:
1) C(x + y) must be the same as C(x) + C(y) and
2) C(cx) must be equal to cC(x).

C(x + y) = B(A(x + y)) = B(A(x) + A(y)) = B(A(x)) + B(A(y)) = C(x) + C(y)
C(cx) = B(A(cx)) = B(cA(x)) = cB(A(x)) = cC(x)

We see that C(x) satisfies both conditions, so it is also a linear transformation.
• At , why doesn't Sal just use the associative property of matrix multiplication to get that (T ∘ S)(x) = B(Ax) = (BA)x, and thus C = BA?
• Well, later he defined matrix multiplication as we know it. So, C is in fact BA.
• In the previous video, the transformation S maps the members from X to Y ( S:X->Y), hence S(x) = B x.

Similarly, The transformation T: Y -> Z, which means, the members of Y to Z. Shouldn't it be ( T(y)=B y ) or T(s(x)) = B s(x)?

why does Sal write T:Y-> Z as T(x) = B x ? and B is matrix of size l x m
• You're right. He'd probably just not being as careful with how he labels thing as maybe he should.
(1 vote)
• How do you determine the size of the matrix from the domain and the codomain? What's the trick here ..?
(1 vote)
• The simple answer is that the matrix will be m x n where the domain is R^n and the codomain is R^m.

The size of a matrix is written m rows by n columns, usually expressed as m x n. For a linear transformation T(x) from R^n (domain) to R^m (codomain) we can express it as a T(x) = A*x, where A is an m x n matrix.

For example a transformation from R^3 to R^2 (e.g. 3D world onto a 2D screen) can be expressed as a 2 x 3 matrix A multiplied by a vector in R^3 which will produce a vector in R^2.