How can I find out if this matrix A's columns are linearly independent?

[shamieh]

How can I find out if this matrix A's columns are linearly independent?

$\begin{bmatrix}1&0\\0&0\end{bmatrix}$

I see here that $x_1 = 0$ and similarly $x_2 = 0$ does this mean that this matrix A's columns are therefore linearly dependent?

Also this is a projection onto the $x_1$ axis so is it also safe to say that this is also one-to-one since it has only the trivial solution of $0$?

 

[Sudharaka]

How can I find out if this matrix A's columns are linearly independent?

$\begin{bmatrix}1&0\\0&0\end{bmatrix}$

I see here that $x_1 = 0$ and similarly $x_2 = 0$ does this mean that this matrix A's columns are therefore linearly dependent?

Also this is a projection onto the $x_1$ axis so is it also safe to say that this is also one-to-one since it has only the trivial solution of $0$?

Hi shamieh, The two columns of the matrix are \(\displaystyle \begin{pmatrix}1\\0\end{pmatrix}\) and \(\displaystyle \begin{pmatrix}0\\0\end{pmatrix}\). To check the linear independence of these two vectors take \(\displaystyle \alpha\) and \(\displaystyle \beta\) such that, ​$\alpha\begin{pmatrix}1\\0\end{pmatrix}+\beta\begin{pmatrix}0\\0\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$.Then, \(\displaystyle \alpha=0\) and \(\displaystyle \beta\in\Re\) is arbitrary. Thus these two vectors are linearly dependent. Another method of finding the linear independence of two vectors is to calculate the determinant of the matrix formed by them. This is illustrated in the following Wikipedia article. https://en.wikipedia.org/wiki/Linear_independence#Alternative_method_using_determinants