Finding the Representing Matrix with Respect to a Basis

This question is within a finite dimension.

Let $u_1=\frac{1}{2}(1,1,-1,-1)$, $u_2=\frac{1}{2}(1,1, 1, 1)$ and let $W=Sp(u_1,u_2)$. Let $W^{\bot}=\{\frac{1}{2}(1,-1,-1,1), \frac{1}{2}(1,-1,1,-1)\}$. $B=\{u_1,u_2,u_3,u_4\}$ is orthonormal basis of $\Bbb R^4$.

Let:

$$T(u_1)=u_2,T(u_2)=-u_1, T(u_3)=0, T(u_4)=0$$.

How do I find the representing matrix $[T]_B$?

I know I should get: $$\begin {bmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 &0 \\ 0 &0&0&0 \end {bmatrix}$$.

How do I get this result?

Thanks,

Alan

• Hint: the columns of the matrix will be the images of the basis vectors expressed in that basis. – amd Feb 29 '16 at 20:21
• Do you know the procedure to get the matrix associated to a linear transformation in specific basis? At what point did you have difficulties? – la flaca Feb 29 '16 at 21:03
• @Eliana exactly when trying to make this process, of getting the transformation "translate" into matrix. – Alan Feb 29 '16 at 21:04

Let $v=a_1 u_1 + a_2 u_2 + a_3 u_3 + a_4 u_4$ and let $coord_B \in Hom(\mathbb{R}^4,\mathbb{R}^4)$ defined as follows: $$coord_B(v)=(a_1,a_2,a_3,a_4)$$ That is, $coord_B$ gives you the coordinates in the basis $B$ of a vector $v \in \mathbb{R}^4$.
So, by definition, $$[T]_B= [coord_B(T(u_1))|coord_B(T(u_2))|coord_B(T(u_3))|coord_B(T(u_4))]$$ And you have that: $$u_1=1u_1+0u_2+0u_3+0u_4$$ $$u_2=0u_1+1u_2+0u_3+0u_4$$ $$u_3=0u_1+0u_2+1u_3+0u_4$$ $$u_4=0u_1+0u_2+0u_3+1u_4$$ So $$coord_B(T(u_1))=coord_B(u_2)=(0,1,0,0)$$ $$coord_B(T(u_2))=coord_B(-u_1)=(-1,0,0,0)$$ $$coord_B(T(u_3))=coord_B(0)=(0,0,0,0)$$ $$coord_B(T(u_4))=coord_B(0)=(0,0,0,0)$$ And there you have your matrix