Linear algebra problem (standard matrix for a linear operator)

[Tsunami317]

Homework Statement

Determine the standard matrix for the linear operator defined by the formula below:
T(x, y, z) = (x-y, y+2z, 2x+y+z)

Homework Equations

The Attempt at a Solution

No idea

 

[HallsofIvy]

I guess the first question is why you have no idea how to do this problem! Are you taking as course in linear algebra? Do you know how to multiply matrices?

Hopefully, at least you know that, because this T is applied to a 3 component vector and the result is a 3 component vector, T will be represented by a 3 by 3 matrix.

And, you should know that applying T to a vector <x, y, z> is the same as the matrix multiplication
$$\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}\begin{bmatrix}x \\ y \\z \end{bmatrix}$$

Because they say the "standard" matrix, they want you to use the "standard" basis, <1, 0, 0>, <0, 1, 0>, and <0, 0, 1>. What do you get when you multiply the matrix above by each of those?

You are told that T(x, y, z) = (x-y, y+2z, 2x+y+z). What is T(1, 0, 0)? What is T(0, 1, 0)? What is T(0, 0, 1)?

Compare those with the result of multiplying the matrix.

 

[Tsunami317]

So the answer would be
[x-y 0 0
0 y+2z 0
0 0 2x+y+z]
?
or is the answer
[x-y
y+2x
2x+y+z]
I understand how to multiply matrices, but now this looks backward to me. Yes I am taking a distance learning class and there is no professor, just me reading a textbook which has terminology that rarely matches up with the sparse classnotes, let alone what I can google, and watching hundreds of hours of Khan Academy.

 

[Tsunami317]

And thank you for your help!

 

[LCKurtz]

Because they say the "standard" matrix, they want you to use the "standard" basis, <1, 0, 0>, <0, 1, 0>, and <0, 0, 1>. What do you get when you multiply the matrix above by each of those?

You are told that T(x, y, z) = (x-y, y+2z, 2x+y+z). What is T(1, 0, 0)? What is T(0, 1, 0)? What is T(0, 0, 1)?

Compare those with the result of multiplying the matrix.

So the answer would be
[x-y 0 0
0 y+2z 0
0 0 2x+y+z]
?
or is the answer
[x-y
y+2x
2x+y+z]

No. It might help if you actually answer the questions Halls asked you above. Those answers might lead you to the solution.

 

[Tsunami317]

I tried to multiply the matrix
1 0 0
0 1 0
0 0 1
by the vectors given but I got this
[x-y
y+2x
2x+y+z]
Doesn't make sense

 

[estro]

HallsofIvy and LCKurtz already provided great hints, I will repeat what they told you in other words:

You need to find a matrix A so that: A(x,y,z)=(x-y, y+2z, 2x+y+z)

It is important to understand the relationship between [linear] transformation and its "representative matrix", to be able to solve this problem.

With all the respect to Khan Academy I don't think it is the right place to learn linear algebra.
Try looking at: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
Prof. Gilbert Stran provides great lectures and excellent book.

 

[LCKurtz]

I tried to multiply the matrix
1 0 0
0 1 0
0 0 1
by the vectors given but I got this
[x-y
y+2x
2x+y+z]
Doesn't make sense

Yes, it doesn't make sense, and it isn't what you were asked to do.

Halls asked you what you get when you multiply the matrix$$
\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$$and the three standard basis vectors which you would write as the matrices$$
\begin{bmatrix}1 \\0 \\ 0\end{bmatrix},\, \begin{bmatrix}0 \\ 1 \\0 \end{bmatrix}
,\, \begin{bmatrix}0 \\ 0 \\1 \end{bmatrix}$$What do you get when you multiply them? That is three different questions with three different answers, and there won't be any $x,y,z$ variables in the answer. Can you do that?