LINEAR ALGEBRA: image of vectors through other basis

[Felafel]

Homework Statement

In $E^3$, given the orthonormal basis B, made of the following vectors $v_1=\frac{1}{\sqrt{2}}(1,1,0); v_2=\frac{1}{\sqrt{2}}(1,-1,0); v_3=(0,0,1)$
and the endomorphism $\phi : E^3 \to E^3$ such that $M^{B,B}_{\phi}$=A where
(1 0 0)
(0 2 0) = A
(0 0 0)
find:
$\phi(e_1), \phi(e_2), \phi(e_3)$ written with respect to the canonical basis
(where e1, e2, e3 are vector of the canonical basis $\mathcal{E}$)

The Attempt at a Solution

here is what I thought, but having no solutions i don't know if it is correct:

I write the vectors of the canonical basis as combination of the vectors of B, also obtaining the $M^{B,\mathcal{E}}$:

(1,0,0)=$a_{11} \cdot \frac{1}{\sqrt{2}}(1,1,0)+ a_{21} \cdot \frac{1}{\sqrt{2}}(1,-1,0)+ a_{31} \cdot (0,0,1)$

(0,1,0)=$a_{12} \cdot \frac{1}{\sqrt{2}}(1,1,0)+ a_{22} \cdot \frac{1}{\sqrt{2}}(1,-1,0)+ a_{32} \cdot (0,0,1)$

(0,0,1)=$a_{13} \cdot \frac{1}{\sqrt{2}}(1,1,0)+ a_{23} \cdot \frac{1}{\sqrt{2}}(1,-1,0)+ a_{33} \cdot (0,0,1)$

getting:

($\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ 0)
($\frac{\sqrt{2}}{2}$ -$\frac{\sqrt{2}}{2}$ 0)= $M^{B, \mathcal{E}}$
( 0 0 1)

so $\phi(e_1), \phi(e_2), \phi(e_3)$ are the columns of this matrix.
Is it correct?
thank you in advance :)

 

[Fredrik]

I haven't thought it all through, but it doesn't look correct to me. I don't see why those columns would contain the $\mathcal E$-components of the $\phi(e_i)$. Do you have an argument for it? Also, I think I have calculated $\phi(e_3)$, and my result is different from yours.

I think the strategy here should be to start with
$$\phi(e_i)=\phi\left( \sum_j M^{B,\mathcal E}_{ji} v_j\right) =\sum_j M^{B,\mathcal E}_{ji}\phi(v_i).$$ Now you just have to compute the $\phi(v_i)$. (Did I understand your definition of $M^{B,\mathcal E}$ correctly?)

 

[Felafel]

i don't think i have understood what you mean :(. could you please do an example of the calculation you have done by using a vector?

 

[Fredrik]

I don't want to give away too much information. We only give hints here, not complete solutions. But I can of course explain what I did there. I just rewrote $e_i$ as a linear combination of the $v_i$ and then I used that $\phi$ is linear.

The $\phi(v_i)$ are easy to calculate, since you've been given the matrix representation of $\phi$ in the $\{v_i\}$ basis.

 

[Felafel]

thank you :) solved it!