Matrices of Linear Maps Review

Matrices of Linear Maps Review

We will now review some of the recent content regarding matrices of linear maps.

• Recall from the Solution Spaces to Systems of Linear Equations page that a system of $m$ linear equations in $n$ unknowns could be represented succinctly as a linear transformation$T \in \mathcal L (\mathbb{F}^n, \mathbb{F}^m)$ as:
(1)
\begin{align} \quad T((x_1, x_2, ..., x_n)) = \left ( \sum_{k=1}^{n} a_{1,k}x_k, \sum_{k=1}^{n} a_{2,k}x_k, ..., \sum_{k=1}^{n} a_{m,k}x_k \right ) \end{align}
• A Homogenous Linear System has the form $T(x_1, x_2, ..., x_n) = 0$ which is equivalent to:
(2)
\begin{align} \sum_{k=1}^{n} a_{1,k}x_k = a_{1,1}x_1 + a_{1,2}x_2 + ... + a_{1,n}x_n = 0 \\ \sum_{k=1}^{n} a_{2,k}x_k = a_{2,1}x_1 + a_{2,2}x_2 + ... + a_{2,n}x_n = 0\\ \vdots \quad \quad \quad \quad\quad \quad\quad \quad \\ \sum_{k=1}^{n} a_{m,k}x_k = a_{m,1}x_1 + a_{m,2}x_2 + ... + a_{m,n}x_n = 0\\ \end{align}
• We already know that homogenous linear systems always have a solution - namely the trivial solution $(x_1, x_2, ..., x_n) = (0, 0, ..., 0)$. However, we proved that if the number of unknowns $n$ is greater than the number of equations $m$ then a nontrivial solution is guaranteed to exist.
• We then looked at Nonhomogenous Linear Systems which have the form $T(x_1, x_2, ..., x_n) = (b_1, b_2, ..., b_m)$ where not all of the $b$'s are equal to zero. This is equivalent to:
(3)
\begin{align} \sum_{k=1}^{n} a_{1,k}x_k = a_{1,1}x_1 + a_{1,2}x_2 + ... + a_{1,n}x_n = b_1 \\ \sum_{k=1}^{n} a_{2,k}x_k = a_{2,1}x_1 + a_{2,2}x_2 + ... + a_{2,n}x_n = b_2\\ \vdots \quad \quad \quad \quad\quad \quad\quad \quad \\ \sum_{k=1}^{n} a_{m,k}x_k = a_{m,1}x_1 + a_{m,2}x_2 + ... + a_{m,n}x_n = b_m\\ \end{align}
• We then proved that if the number of equations $m$ is greater than the number of unknowns $n$ then there exists a vector $b = (b_1, b_2, ..., b_m)$ for which there exists no solution to the nonhomogenous linear system from above.
• On the The Matrix of a Linear Map we said that if $V$ and $W$ are both finite-dimensional vector spaces with bases $B_V = \{ v_1, v_2, ..., v_n \}$ and $B_W = \{ w_1, w_2, ..., w_m \}$ respectively and if $T \in \mathcal L (V)$ then the Matrix of $T$ with respect to the bases $B_V$ and $B_W$ is denoted $\mathcal M (T, B_V, B_W)$ where $T(v_k) = a_{1,k}w_1 + a_{2,k}w_2 + ... + a_{m, k} w_m$ for each $k = 1, 2, ..., n$ is given by:
(4)
\begin{align} \quad \mathcal M (T, B_V, B_W) = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix} \end{align}
• In other words, the matrix of $T$ with respect to $B_V$ and $B_W$ can be obtained by applying the linear map $T$ to each basis vector. The columns of $\mathcal M (T, B_V, B_W)$ will be the values of the coefficients of the linear combination of the basis vectors from $W$ which equal to $T$ when applied to basis vectors in $V$.