Question

Let S be a linear transformation from R 3 to R 2 with associated matrix A = [ −2 1 0 −3 5 3 ] .

Answer 1

The matrix C of the composition T∘S is:
`C=\left[\begin{array}{ccc}-6\\-4\\9\end{array}\right]`

To find the matrix C of the composition T∘S, where S is a linear transformation from R3 to R2 and T is a linear transformation from R2 to, we use the matrix multiplication.

Given the matrices:

`A=\left[\begin{array}{ccc}2&1&-2\\-3&-2&-3\\\end{array}\right]`

and

`B=\left[\begin{array}{ccc}3&-3\\22&0\\\end{array}\right]`

The matrix C is obtained by multiplying matrix B with matrix A:

C=B⋅A

Performing the matrix multiplication:

`C=\left[\begin{array}{ccc}2&1&-2\\-3&-2&-3\\\end{array}\right].\left[\begin{array}{ccc}3&-3\\22&0\\\end{array}\right]`

C=[ (−2⋅3+1⋅(−3)+(−2)⋅(−3))(22⋅3+0⋅(−3))​ (−2⋅(−2)+1⋅(−2)+(−2)⋅(−3)(22⋅(−2)+0⋅(−2)) ]

`C=\left[\begin{array}{ccc}-6&-4\\66&-44\\\end{array}\right]`

Therefore, the matrix C is:

`C=\left[\begin{array}{ccc}-6\\-4\\9\end{array}\right]`