How do you show $\text{col}A= \text{span}(c)$ and $\text{row }A= \text{span}(r)$ based on the following condition.

Question

Let $A=cR$ where $c \ne 0$ is a column in $ℝ^m$ and $R \ne 0$ is a row in $ℝ^n$. Prove $\text{col}A= \text{span}(c)$ and $\text{row }A= \text{span}(R)$.

Could you give me an approach?

Answer 1

Hint: Write $c = (c_1, \ldots, c_m)^T$ and $R = (r_1, \ldots, r_n)$. Then

$$ cR = \begin{pmatrix} c_1r_1 & \ldots & c_1 r_n \\ c_2r_1 & \ldots & c_2 r_n \\ \vdots & \ldots & \vdots \\ c_mr_1 & \ldots & c_mr_n \end{pmatrix}. $$