Is set of vectors dependent in R²

Question

Given a set of vectors: {$(-1,2),(2,-4)$}

How can we say that this set of vectors dependent in $\mathbb{R}^2$?

On solving: $$a(X_1)+b(X_2)=0$$

$a,b=0$, shouldn't it be independent? But answer says that it's dependent.

Answer 1

The vectors are linearly independent if the only possible solution to $aX_1 + bX_2 = 0$ is $ a = b = 0$. Notice that no matter what $X_1$ and $X_2$ are, $a = b = 0$ always gives you a linear combination that sums to zero. That is why choosing scalars that are all zero is referred to as the "trivial" solution to $aX_1 + bX_2 = 0$. If there is a non-trivial solution to $aX_1 + bX_2 = 0$, then $X_1$ and $X_2$ are linearly dependent. Notice that in your case $a = -2$ and $b = 1$ are non-trivial choices for $a$ and $b$ that satisfy $aX_1 + bX_2 = 0$.

Answer 2

As noted in the comment , a simple inspection shows that $X_2=(2,-4)^T=-2(-1,2)^T=-2 X_1$. But, if you want to show that $aX_1+bX_2=0$ for $a,b$ not null, you have to interpret correctly this equation that is (in components) $$ a\begin{pmatrix} -1\\2 \end{pmatrix}+b \begin{pmatrix} 2\\-4 \end{pmatrix} =\begin{pmatrix} 0\\0 \end{pmatrix} $$ that is equivalent to the linear system $$ \begin{cases} -a+2b=0\\ 2a-4b=0 \end{cases} $$

Can you solve it for $a,b$? And show that we have many solutions ?