04. Linear Combination -Quiz 1

The following is a set of three vectors:

(1) \vec{v_1}=\begin{bmatrix} 1\\ 2\\ 3\end{bmatrix}

(2)
\vec{v_2}=\begin{bmatrix} 2\\ 2\\ 2\end{bmatrix}

(3) \vec{v_3}=\begin{bmatrix} 8\\ 8\\ 8\end{bmatrix}

All three vectors are \in\mathbb{R^3}

The question in this quiz may seem a bit strange. We have three vectors, why do they not all define the plane that can be spanned by a linear combination of them all?

A simple glance at vectors \vec{v_2} and \vec{v_3} will show you that one vector can be defined as a linear combination of the other.

for example:

\vec{v_2} =0.25\vec{v_3}

\begin{bmatrix} 2\\ 2\\ 2\end{bmatrix}=0.25\begin{bmatrix} 8\\ 8\\ 8\end{bmatrix}

In other words, if we use \vec{v_2} as a part of our linear combination (for creating finding the vectors spanned), we do not need \vec{v_3}. And vice versa: if we use \vec{v_3} as a part of our linear combination (for creating finding the vectors spanned), we do not need \vec{v_2}.

Therefore, to define the plane spanned by a linear combination of the vectors above, we need (\vec{v_2} and \vec{v_1}) or (\vec{v_3} and \vec{v_1}).