What is the matrix representation of T ο L with respect to E?

[crashh]

Homework Statement

The problem states that we have L as the linear transformation as:
\begin{align*}
A=
\left(
\begin{array}{ccc}
2 & 0 & 1 \\
-2 & 3 & 2 \\
4 & 1 & 5
\end{array}
\right)
\end{align*}

And when given another linear transformation T as:
\begin{align*}
B=
\left(
\begin{array}{ccc}
-3 & 1 & 0 \\
2 & 0 & 1 \\
0 & -1 & 3
\end{array}
\right)
\end{align*}

Then find the matrix representation of T ο L with respect to E(which is the standard basis, as are both transformations).

T ο L is the composition of T and L.

Homework Equations

I assumed you could just multiply the two matrices togeather, as they share the same basis, thus getting the composition of the two lineartransformations?

The Attempt at a Solution

\begin{align*}
T&=B=
\left(
\begin{array}{ccc}
-3 & 1 & 0 \\
2 & 0 & 1 \\
0 & -1 & 3
\end{array}
\right)\\
L&=A=
\left(
\begin{array}{ccc}
2 & 0 & 1 \\
-2 & 3 & 2 \\
4 & 1 & 5
\end{array}
\right)\\
\end{align*}

\begin{align*}
BA&=
\left(
\begin{array}{ccc}
-3 & 1 & 0 \\
2 & 0 & 1 \\
0 & -1 & 3
\end{array}
\right)
\left(
\begin{array}{ccc}
2 & 0 & 1 \\
-2 & 3 & 2 \\
4 & 1 & 5
\end{array}
\right)\\
&=
\left(
\begin{array}{ccc}
-8 & 3 & -1 \\
8 & 1 & 7 \\
14 & 0 & 13
\end{array}
\right)
\end{align*}

Or is this completely wrong?

 

[vela]

I didn't check that you multiplied the matrices together correctly, but assuming you did, that's the right answer.

 

[Fredrik]

I assumed you could just multiply the two matrices togeather, as they share the same basis, thus getting the composition of the two lineartransformations?

No need to assume it. The number on row i, column j of the matrix corresponding to an arbitrary linear operator A is $A_{ij}=\langle e_i,Ae_j\rangle$. So
\begin{align}
&(T\circ L)_{ij}=\langle e_i,(T\circ L)e_j\rangle =\langle e_i,T(Le_j)\rangle =\left\langle e_i,T\left(\sum_k \langle e_k,Le_j\rangle e_k\right)\right\rangle\\
&=\sum_k\langle e_k,Le_j\rangle \langle e_i,Te_k\rangle =\sum_k L_{kj}T_{ik} =\sum_k T_{ik}L_{kj}.
\end{align}

 

[HallsofIvy]

Yes, that is the correct product of the two matrices. One way to remember how to multiply matrices is to think of the rows of the first matrix and the columns of the second matrix as "vectors". If a_{ij} is the number in the i row, j column of the product matrix, then a_{ij} is the dot product of the i row of the first matrix and the j column of the second matrix.

For example, the first row of the first matrix, B, is <-3, 1, 0> and the first column of the second matrix, A, is <2, -, 4> so the number in the first row, first column, of BA is (-3)(2)+ 1(-2)+ 0(4)= -8 as you have.

And, of course, the order of the multiplication (BA rather than AB) is important because matrix multiplication is NOT commutative. The multiplication of the matrices must be the same as the order in which the transformation are applied.