Question

Let `\mathcal{E} = {[[1],[0]][[0],[1]]}` and `\mathcal{B} = {[[3],[1]][[-2],[1]]}` The vector `vecv` relative to `\mathcal{B}` is `[vecv]_\mathcal{B}= [[2],[1]]` . Find `vecv` relative to `\mathcal{E}` `[vecv]_\mathcal{B}`?

Answer 1

The answer is `=((4),(3))`

Explanation:

The canonical basis is `E={((1),(0)),((0),(1))}`

The other basis is `B={((3),(1)),((-2),(1))}`

The matrix of change of basis from `B` to `E` is

`P=((3,-2),(1,1))`

The vector `[v]_B=((2),(1))` relative to the basis `B` has coordinates

`[v]_E=((3,-2),(1,1))((2),(1))=((4),(3))`

relative to the basis `E`

Verification :

`P^-1=((1/5,2/5),(-1/5,3/5))`

Therefore,

`[v]_B=((1/5,2/5),(-1/5,3/5))((4),(3))=((2),(1))`