1. ## Matrix 2x2

If A = [a b]
c d
is a 2 x 2 matrix such that the vectors [a c] and [b d]

are linearly independent, prove carefully that rank A = 2. (You cannot choose thematrix A - your proof must work for every 2 2 matrix with the property above,i.e. every 2 2 matrix with independent columns.)

2. ## Re: Matrix 2x2

Essentially, you want to show that $\displaystyle \begin{bmatrix}a \\ c\end{bmatrix}$ and $\displaystyle \begin{bmatrix}b \\ d\end{bmatrix}$ are linearly independent if and only if $\displaystyle \begin{bmatrix}a & b\end{bmatrix}$ and $\displaystyle \begin{bmatrix}c & d\end{bmatrix}$ are linearly independent.

3. ## Re: Matrix 2x2

yea that is right. but how to do that?

4. ## Re: Matrix 2x2

Assume that any linear combination $\displaystyle e\begin{bmatrix}a \\ c\end{bmatrix}+f\begin{bmatrix}b \\ d\end{bmatrix} \neq \begin{bmatrix}0 \\ 0\end{bmatrix}$ unless $\displaystyle e=f=0$. Then show that any linear combination $\displaystyle e\begin{bmatrix}a & b\end{bmatrix}+f\begin{bmatrix}c & d\end{bmatrix} \neq \begin{bmatrix}0 & 0\end{bmatrix}$ unless $\displaystyle e=f=0$.