Solution 3d 20

Suppose $n$ is a positive integer and $A_{i,j} \in \mathbf F$ for $i,j = 1,\dots,n$ . Prove that the following are equivalent (note that in both parts below, the number of equations equals the number of variables):

The trivial solution $x_1 = \dots = x_n = 0$ is the only solution to the homogeneous system of equations

\begin{aligned} \sum_{k=1}^n A_{1,k}x_k &= 0 \\ &\vdots \\ \sum_{k=1}^n A_{n,k}x_k &= 0 \\ \end{aligned}

For every $c_1,\dots,c_n \in \mathbf F$ , there exists a solution to the system of equations

\begin{aligned} \sum_{k=1}^n A_{1,k}x_k &= 0 \\ &\vdots \\ \sum_{k=1}^n A_{n,k}x_k &= 0 \\ \end{aligned}

Let $T \in \mathcal L(\mathbf F^n)$ be the linear operator corresponding to the matrix $A$ . 3.69 tells the following are equvalent

$Tx = 0$ iff $x = 0$ ( $T$ is injective)
for every $b \in \mathbf F^n$ there exists an $x \in \mathbf F^n$ with $Tx = b$ ( $T$ is surjective)

todo: cite theorems to make more rigorous (it's obvious to me so I'm not motivated to be rigorous)