Matrix for a linear map

change-of-basislinear algebralinear-transformationsvector-spaces

Consider $V= \left \{ v= \begin{bmatrix}
x_1\\
x_2\\
x_3
\end{bmatrix} : x_1-x_2+2x_3=0\right \}\subset K^3$ and the linear map $f:V \rightarrow K^2$ defined by

$f\begin{pmatrix}
x_1\\
x_2\\
x_3
\end{pmatrix}=\begin{pmatrix}
x_1-x_3\\
x_2-3x_3
\end{pmatrix}$

Now, what is the matrix $M$ that represents $f:V \rightarrow K^2$ with respect to the basis of $V$ and the standard basis $(e_1,e_2)$ for $K^2$.

So far

I found the basis of V of dimension 2 to be the vectors ${(1, 1, 0), (-2, 0, 1)}$. But then I do not know how to continue. Maybe coordinate shift?

Best Answer

There are multiple errors/unclear parts in your question. I'm not very sure what the set $K$ is, so I will be assuming it's the field $\mathbb{R}$.

Also, the codomain of $f$ seems to be $K^2$ rather than $K^3$, so I assume $f:V \to K^3$ was a typographic error on your part.

Now, a "matrix representation of a linear transformation $g:A \to B$" only makes sense when we're given specific ordered bases of $A$ and $B$. You gave us $B_{K_2}=(e_1,e_2)$ for $K^2$ but ordered the basis of $V$ with respect to which we will find the matrix representation is not very clear. I will assume it is the ordered basis $B_{V}=((1,1,0),(-2,0,1))$.

I will assume that you are familiar with the theory behind matrix representations of a linear transformation and I will simply find the representation. If anything within this process is unclear, just let me know.

First, we will find the images of elements of $B_V$ under $f$, expressed as linear combinations of elements of $B_{K^2}$.

$$f(1,1,0)=(1-0,1-3 \cdot 0)=(1,1)=1 \cdot (1,0) + 1\cdot (0,1) = \textbf{1} \cdot e_1 + \textbf{1} \cdot e_2$$ $$f(-2,0,1)=(-2-1,0-3\cdot 1)=(-3,-3)=(-3) \cdot(1,0) + (-3) \cdot (0,1) = \textbf{(-3)} \cdot e_1 + \textbf{(-3)} \cdot e_2$$

Now, we will simply take the coefficients written in bold and place them in our representation matrix. Remember, the i-th column represents the coefficients of the linear combination of the i-th element of $B_V$ in terms of the elements of $B_{K^2}$.

$$[f]^{B_V}_{B_{K^2}}= \begin{bmatrix} 1 & -3 \\ 1 & -3 \end{bmatrix}$$

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