I have the following vectors and need to determine if they are linearly independent/dependent. I know that for linear independence, the coefficients multiplied by each vector must equal zero, however, I am confused as to how I solve for that in 2×2 matrices.
My guess is I do C1(A) + C2(B) + C3(C) = 0.
Then from there, I need to prove that C1,C2,C3 (scalar constants) = 0.
My problem is I am not sure about how I go about doing that.
A=
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}
B=
\begin{pmatrix}
0& 1\\
0& 0
\end{pmatrix}
C=
\begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}
Best Answer
If you set $a A + b B +c C = 0$, then computing the matrix gives $a A + b B +c C = \begin{bmatrix} a & b \\ c & a\end{bmatrix} = 0$. It follows that we must have $a=b=c= 0$, hence $A,B,C$ are linearly independent.