# Linear transformation help needed

1. Jun 18, 2009

### zhfs

1. The problem statement, all variables and given/known data

let T : R3 --> R2 be the transformation defined by T([x]) = [y+z]
y x+z
z

(a) show that T is a linear transformation

(b) calculate [T]B',B- the matrix of T with respect to the bases B and B' where
1 0 0
B = { [0] [1] [0] } and B' = {[1] [-1]}
0 0 1 1 1
1
(c) determine the coordinates of T{[1]} with respect to basis B'
1
2. Relevant equations

nothing much can help...

3. The attempt at a solution

part (a) is fine, can be proved

but have little problem in part (b)

since B is 3*3 matrix and B' is 2*2, i can't keep going on that

i've got [T]B={[0] [1][1]}
[1] [0][1]

and i can't keep going from here

and same as part (c) , how can i transfer a 3*1 into with respect 2*2

2. Jun 18, 2009

### zhfs

i don't know whats wrong with my typing

i tis not showing right

but B is standard basis for R3 100,010,001

and B' is [1 -1; 1 1]

3. Jun 18, 2009

### HallsofIvy

Staff Emeritus
Use LaTex! Web sites in general do not respect spaces.

The linear transformation, T, is given by
$$T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right)= \begin{bmatrix} y+ z \\ x+ z\end{bmatrix}$$
(Click on the formula to see the code.)

To show that this is a linear transformation, you must show that T(u+ v)= T(u)+ T(v) for any u, v in R3 and that T(au)= aT(u) for any u in R3 and any real number a.

If T:U-> V and you are given bases for U and V, to write T as a matrix do this:
Apply T to each of the basis vector of U in turn. Write the result of each as a linear combination of the basis vector for V. The coefficients of each linear combination is a column in the matrix. For example
$$T\left(\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\right)= \begin{bmatrix}1 \\ 1\end{bmatrix}= 0\begin{bmatrix}1 \\ -1\end{bmatrix}+ 1\begin{bmatrix}1 \\ 1\end{bmatrix}$$
so the third column of the matrix is
$$\begin{bmatrix}0 \\ 1\end{bmatrix}$$

Note that the order in which the basis vectors are given is important. [0 0 1] is the third vector in the basis for R3 so applying T to it gives the third column of the matrix.

Last edited: Jun 18, 2009