How Do You Find the Standard Matrix Representation for a Linear Transformation?

[DanielFaraday]

Homework Statement

Write the standard matrix representation for T₁ and use it to find [T₁(1,-3,0)]_E.

Homework Equations

<br /> T_1\left(x_1,x_2,x_3\right)=\left(x_3,-x_1,x_3\right)<br />

The Attempt at a Solution

I just wanted to check to see if I am doing this right. Thanks in advance!

<br /> A=\left(<br /> \begin{array}{ccc}<br /> 0 &amp; 0 &amp; 1 \\<br /> -1 &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1<br /> \end{array}<br /> \right)\<br />

<br /> \left[T_1(1,-3,0)\right]_E=A\left(<br /> \begin{array}{c}<br /> 1 \\<br /> -3 \\<br /> 0<br /> \end{array}<br /> \right)=\left(<br /> \begin{array}{ccc}<br /> 0 &amp; 0 &amp; 1 \\<br /> -1 &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1<br /> \end{array}<br /> \right).\left(<br /> \begin{array}{c}<br /> 1 \\<br /> -3 \\<br /> 0<br /> \end{array}<br /> \right)=\left(<br /> \begin{array}{c}<br /> 0 \\<br /> -1 \\<br /> 0<br /> \end{array}<br /> \right)<br />

 

[Office_Shredder]

Your A is transposed from what it should be.

 

[HallsofIvy]

No, Officeshredder,
Applying T₁ to each basis vector in turn gives the columns.

T₁(1, 0, 0)= (0, -1, 0)
T₁(0, 1, 0)= (0, 0, 0)
T₁(0, 0, 1)= (1, 0, 1)

So T₁ is represented by
\begin{bmatrix} 0 &amp; 0 &amp; 1 \\ -1 &amp; 0 &amp; 0 \\ 1 &amp; 0 &amp; 1 \end{bmatrix}
exactly what DanielFaraday has.

And, of course, T₁(1,-3,0)= (0,-1,0) as said.

 

[DanielFaraday]

Thank you both for your input!

 

[HallsofIvy]

Office shredder may be using a different convention than you and I:

T_1(1,-3,0)= \begin{bmatrix}1 &amp; -3 &amp; 0\end{bmatrix}\begin{bmatrix}1 &amp; -1 &amp; 0\\ 0 &amp; 0 &amp; 0 \\ 1 &amp; 0 &amp; 1\end{bmatrix}= \begin{bmatrix}0 \\ -1 \\ 0\end{bmatrix}

 

[DanielFaraday]

Yes, these things often depend on the textbook. Thanks.

 

[Office_Shredder]

No, sorry, that was just a brain fart on my part