If $T: V \to W$ where $V \subset R^n$ and $W \subset R^m$ has basis $\beta_1=\{v_1 \dots v_k \}$ and $\Omega_1=\{ w_1 \dots w_l\}$ respectively (thus $k < n$ and $l < m$).
-
Can I represent this by $T(x)=A[x]_{\beta_1}$, where $a_{ij}$ is found from $T(v_j)=\sum_{i= 1}^l a_{ij}w_i$ and $[x]_{\beta_1} =\begin{bmatrix}c_1\\\vdots\\c_k\end{bmatrix}\$ $ from $x = \sum_{i=1}^k c_iv_i$? I'm trying to wrap my head around the fact that this coordinate representation of $x$ is a $k$-tuple, but the actual vector $x\in V \subset R^n$ has n-components (and likewise $A[x]_{\beta_1}$ are $l<m$ tuples). Essentially, when using matrices to describe a linear map, the input / output are coordinates relative to some basis with number of coordinates equalling the dimension of the domain / target space? Only when $V=R^n$ are the coordinates of $x$ an $n$-tuple?
-
For $V$, if I have a second basis $\beta_2=\{v_1' \dots v_k' \}$, is the change of basis matrix $k\times k$ that takes $k$-tuple coordinates from one basis to another?
-
What vectors are in the column, row, null, and left null space of $A$, given $A$ is $l \times k$ but the vectors in the domain and image are coordinate representation of vectors in $R^n$ and $R^m$ respectively?
Best Answer
$1.$ Assuming $A$ is the matrix representation of $\textsf T$ with respect to the basis $\beta_1$ and $\Omega_1$, you have a little mistake, is $$[\textsf T(x)]_{\Omega_1} = A[x]_{\beta_1}$$ instead of $\textsf T (x)=A[x]_{\beta_1}$.
And for your existential doubt, consider the subspace $\textsf W = \{ (x_1,x_2,0):\, x_1,x_2\in \mathbb R \}$ of $\mathbb R^3$ and $\gamma = \{ (1,0,0),(0,1,0) \}$ one of its basis.
Then, for any vector $x=(x_1,x_2,0)$ in $\textsf W$, we have $$[(x_1,x_2,0)]_\gamma = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$ since $(x_1,x_2,0)=x_1(1,0,0)+x_2(0,1,0)$.
Yes, $x$ is $3$-tuple and its coordinate representation is a $2$-tuple.
$2.$ That's right, let's say we have a matrix $Q$ so that its $j$-th column is $[v_j]_{\beta_2}$, then $Q$ is the matrix that changes $\beta_1$-coordinates into $\beta_2$-coordinates : $$Q[x]_{\beta_1} = [x]_{\beta_2}$$