Explain why the columns of an $n \times n$ matrix $A$ span $\Bbb R^n$ when $A$ is invertible

# Explain why the columns of an $n \times n$ matrix $A$ span $\Bbb R^n$ when $A$ is invertible

J
240 points

Explain why the columns of an $n \times n$ matrix $A$ span $\Bbb R^n$ when $A$ is invertible. [Hint: Review Theorem 4 in Section 1.4.]

Explain why
jeffp

8.7k points

If $A$ is invertible then $Ax=b \longrightarrow x=A^{-1}b$ is a solution for every $b$ in $\Bbb R^n$ which implies that the columns of $A$ span $\Bbb R^n$

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