I've come across the question:
Prove that if $u$ can be written as a linear combination of $v$ and $w$ in (at least) two different ways, then $v$ and $w$ are linearly dependent.
Based on my knowledge, two vectors are linearly dependent if one is the subset (or a multiple) of the other.
And from the question, I can obtain the following: (I'm not quite sure about the second one): $$u = av + bw$$
$$ u = cw+dv$$
Using the above equations, I don't see how we can prove the linear dependency of $v$ and $w$. Any help is appreciated!
Best Answer
Hint:
Two vectors, $\textbf{v}_1, \textbf{v}_2$ are linearly dependent if there are coefficients $a_1,a_2$, not all zero, such that $a_1\textbf{v}_1 + a_2\textbf{v}_2$ is equal to the zero vector.
From the two equations you've obtained, try to obtain an equation in the form of the equation $a_1\textbf{v} + a_2\textbf{w} = \textbf{0}$