The components of all linear combinations

Question

My linear Algebra homework has this question on it and I was wondering what the answer is.

The components of all linear combinations of v = (1, -2, 1) and w = (0, 1, -1) add up to _____

A simple google search came up with nothing so I thought I'd ask here.

If I had to guess I would say that they add up to 0 because by definition a linear combination is what you get when you resize any of the vectors by a skalar and add the two together, so I thought if you add all of linear combinations you'd get the span, and since the span is an infinite plane in all directions for these particular vectors, all numbers have a corresponding negative. So the sum would be 0.

Answer 1

Note that $\sum_{k=1}^3 [\alpha v + \beta w]_k = \alpha (\sum_{k=1}^3 [v]_k) + \beta (\sum_{k=1}^3 [w]_k) $.

Now compute $\sum_{k=1}^3 [v]_k = ?$ and $\sum_{k=1}^3 [w]_k = ?$.

Answer 2

A linear combination has the form $$ \alpha(1,-2,1)+\beta(0,1,-1)= (\alpha,-2\alpha+\beta,\alpha-\beta) $$ Then $$ \alpha+(-2\alpha+\beta)+(\alpha-\beta)=0 $$

A nicer way to show this is to write vectors as columns and notice that a linear combination of the two vectors is $$ \begin{bmatrix} 1 & 0 \\ -2 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} $$ and that the sum of the components of the vector $v=\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}$ is $$ \begin{bmatrix} 1 & 1 & 1 \end{bmatrix} \begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix} =v_1+v_2+v_3 $$ Since $$ \begin{bmatrix} 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -2 & 1 \\ 1 & -1 \end{bmatrix} =\begin{bmatrix}0 & 0\end{bmatrix} $$ we also have $$ \begin{bmatrix} 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -2 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} = \begin{bmatrix}0 & 0\end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} =0 $$