Let $V=\mathbb{R}^4$, let $W=\mathbb{R}^3$ and let $\phi$ be the
linear map $$\left(x,y,z,t\right)\rightarrow
\left(x-2z+t,2y+z,x+4y+t\right)$$Write down the matrix of A of $\phi$ with respect to the standard
bases of $\mathbb{R}^4$ and $\mathbb{R}^3$.
Could someone please help me with the method for finding the matrix A.
I understand we use the standard bases of $\mathbb{R}^4$:
$$e_1=\left(1,0,0,0\right), e_2=\left(0,1,0,0\right), e_4=\left(0,0,1,0\right), e_4=\left(0,0,0,1\right)$$
and the standard bases of $\mathbb{R}^3$:
$$f_1=\left(1,0,0\right), f_2=\left(0,1,0\right), f_3=\left(0,0,1\right)$$
But I'm not sure how.
Best Answer
First, we need to check where our basis for $\mathbb{R}^4$ is sent. $$e_1\mapsto (1,0,1)=f_1+f_3$$ $$e_2\mapsto (0,2,0)=2f_2+4f_3$$ $$e_3\mapsto (-2,1,0)=-2f_1+f_2$$ $$e_4\mapsto (1,0,1)=f_1+f_3$$ So, using the matrix "algorithm" $$ A=\begin{bmatrix} 1&0&-2&1\\ 0&2&1&0\\ 1&4&0&1\\ \end{bmatrix}.$$
More precisely: given a linear transformation $T:V\to W$, where $V$ and $W$ are abstract vector spaces (over $\mathbb{F}$) with bases $v_1,\ldots,v_n$ and $w_1,\ldots,w_m$ respectively, the $i^{th}$ column of the matrix representing $T$ with respect to these bases is
$$ \begin{pmatrix} a_{1,i}\\ \vdots\\ a_{m,i} \end{pmatrix}.$$ Here the $a_{j,i}\in\mathbb{F}$ are the coefficients satisfying $$ T(v_i)=\sum_{k=1}^m a_{k,i}w_k.$$