# Linear Transformation from R3 to R3

• jolly_math

#### jolly_math

Homework Statement
Describe explicitly a linear transformation from R3 into R3 which has as its
range the subspace spanned by (1, 0, -1) and (1, 2, 2).
Relevant Equations
linear transformation
"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?

Thank you.

Does it help to glance at the following matrices:
$$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 2 \\ -1 & -1 & 2\\ \end{bmatrix} \times \begin{bmatrix} x \\ y \\ z\\ \end{bmatrix}$$
?

• jolly_math
jolly_math said:
Homework Statement:: Describe explicitly a linear transformation from R3 into R3 which has as its
range the subspace spanned by (1, 0, -1) and (1, 2, 2).
Relevant Equations:: linear transformation

"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?

Thank you.
A linear transformation can be fully described by its action on any basis. Can you see why?

• jolly_math
Hall said:
$$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 2 \\ -1 & -1 & 2\\ \end{bmatrix} \times \begin{bmatrix} x \\ y \\ z\\ \end{bmatrix}$$

For R3, would
$$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 0 \\ -1 & 2 & -1\\ \end{bmatrix} \times \begin{bmatrix} x \\ y \\ z\\ \end{bmatrix}$$
also work (switching which vector corresponds to each basis)? Thanks.

jolly_math said:
For R3, would
$$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 0 \\ -1 & 2 & -1\\ \end{bmatrix} \times \begin{bmatrix} x \\ y \\ z\\ \end{bmatrix}$$
also work (switching which vector corresponds to each basis)? Thanks.
Yes, I think. I think even double columns of ##(1,2,2)## will also satisfy the given things.

• jolly_math