Question

Question: Consider an $\mathit{n}\mathbf{\times }\mathit{p}$matrix Aand a$\mathit{p}\mathbf{\times }\mathit{m}$matrix B. We are told that the columns of A and the columns of B are linearly independent. Are the columns of the product AB linearly independent as well?

Short answer

If we have an$n\times p$matrix A and a$p\times m$matrix B and we are told that the columns of A and the columns of B are linearly independent then the columns of the product AB are linearly independent as well.

Step-by-step solution

 Mentioning the concept

We have an $n\times p$ matrix A and a $p\times m$ matrix B such that

localid="1659355364118" $A=\left[{\overrightarrow{u}}_1,{\overrightarrow{u}}_2...{\overrightarrow{u}}_p\right]$and$B=\left[{\overrightarrow{v}}_1,{\overrightarrow{v}}_2...{\overrightarrow{v}}_m\right]$

Since the columns of B are linearly independent, then there exists scalars$a_1,a_2,...a_m,$,

all of them zero, such that

$a_1{\overrightarrow{v}}_1+a_2{\overrightarrow{v}}_2+....+a_m{\overrightarrow{v}}_m=0$ (1)

To prove the the columns of the product AB linearly independent as well

Note that the matrixAB is formed as the matrix:

$$\begin{gathered} AB=A\left[{\overrightarrow{v}}_1,{\overrightarrow{v}}_2...{\overrightarrow{v}}_m\right] \\ \Rightarrow AB=\left[A{\overrightarrow{v}}_1,A{\overrightarrow{v}}_2...A{\overrightarrow{v}}_m\right] \end{gathered}$$

Thus, we have the vectors$A{\overrightarrow{v}}_1,A{\overrightarrow{v}}_2...A{\overrightarrow{v}}_m$as the columns of the matrixAB.

Now, if we multiply the equation (1) by A from left we get,

$a_{1}A{\overrightarrow{v}}_1,a_{2}A{\overrightarrow{v}}_2...a_{m}A{\overrightarrow{v}}_m$

Since the scalars$a_1,a_2,...a_m$are all zero, therefore the vectors$A{\overrightarrow{v}}_1,A{\overrightarrow{v}}_2...A{\overrightarrow{v}}_m$ are linearly independent.

Since the vectors$A{\overrightarrow{v}}_1,A{\overrightarrow{v}}_2...A{\overrightarrow{v}}_m$are the columns of the matrix A, thusthe columns of the product AB are linearly independent as well.

Final Answer

If we have an$n\times p$matrix A and a$p\times m$matrix B and we are told that the columns of A and the columns of B are linearly independent, then the product AB columns are linearly independent.