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Suppose $u_{1}, u_{2}, \ldots, u_{n}$ belong to a vector space $V$ over a field $K,$ and suppose $P=\left[a_{i j}\right]$ is an $n$ -square matrix over $K .$ For $i...

Question

Suppose $u_{1}, u_{2}, \ldots, u_{n}$ belong to a vector space $V$ over a field $K,$ and suppose $P=\left[a_{i j}\right]$ is an $n$ -square matrix over $K .$ For $i=1,2, \ldots, n,$ let $v_{i}=a_{i 1} u_{1}+a_{i 2} u_{2}+\cdots+a_{i n} u_{n}$ (a) Suppose $P$ is invertible. Show that $\left\{u_{i}\right\}$ and $\left\{v_{i}\right\}$ span the same subspace of $V$. Hence, $\left\{u_{i}\right\}$ is linearly independent if and only if $\left\{v_{i}\right\}$ is linearly independent. (b) Suppose $P$ is

Suppose $u_{1}, u_{2}, \ldots, u_{n}$ belong to a vector space $V$ over a field $K,$ and suppose $P=\left[a_{i j}\right]$ is an $n$ -square matrix over $K .$ For $i=1,2, \ldots, n,$ let $v_{i}=a_{i 1} u_{1}+a_{i 2} u_{2}+\cdots+a_{i n} u_{n}$ (a) Suppose $P$ is invertible. Show that $\left\{u_{i}\right\}$ and $\left\{v_{i}\right\}$ span the same subspace of $V$. Hence, $\left\{u_{i}\right\}$ is linearly independent if and only if $\left\{v_{i}\right\}$ is linearly independent. (b) Suppose $P$ is singular (not invertible). Show that $\left\{v_{i}\right\}$ is linearly dependent. (c) Suppose $\left\{v_{i}\right\}$ is linearly independent. Show that $P$ is invertible.



Answers

Suppose the columns of a matrix $A=\left[\begin{array}{lll}{a_{1}} & {\cdots} & {a_{p}}\end{array}\right]$ are linearly independent. Explain why $\left\{\mathbf{a}_{1}, \ldots, \mathbf{a}_{p}\right\}$ is a basis for
Col $A .$


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