Find $ \ [v]_B \ $ and interpret what it means.

Question

Let $ B=\{(1,0,1), \ (1,1,-1) , \ (2,0,1) \} \ $ be a basis for $ \ \mathbb{R}^3 \ $ and let $ \ v=(-9,1,-8) \ $.

Find $ \ [v]_B \ $ and interpret what it means.

Answer:

My question is if the transformation is not given then how to find the matrix $ \ [v]_B \ $ ?

Answer 1

Let $E$ be the canonical basis $\{(1,0,0), (0,1,0), (0,0,1) \} $

The columns of the matrix $$P=\begin{bmatrix} 1&1&2 \\0&1&0 \\ 1&-1&1 \end{bmatrix} $$ are formed by where the basis vectors are mapped to in the new basis. $B=\{(1,0,1), (1,1,-1), (2,0,1) \} $

I assume that $(-9,1,-8) $ are the coordinates of a point with respect to the base $E$

To find the coordinates of that point with respect to the new basis we need $P^{-1} $ $$P^{-1} =\begin{bmatrix} -1&3&2 \\0&1&0\\ 1&-2&-1 \end{bmatrix} $$

Finally $$[v]_B=\begin{bmatrix} -1&3&2 \\0&1&0\\ 1&-2&-1 \end{bmatrix} \begin{bmatrix} -9 \\1 \\ -8 \end{bmatrix}=\begin{bmatrix} -4 \\1 \\ -3 \end{bmatrix}$$

Answer 2

Let A be a matrix where each column is one of the base vectors. Now you are solving Ax=v. You are probably supposed to understand that you are then using a new set of coordinates. (basis ~ coordinates)