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# How to compare two spans?

## Problem

Let $\bx$, $\by$, $\bz$, and $\bw$ be vectors in $\mathbb{R}^n$ such that $\bw = \by + \bz$. Show that
$$
\vspan(\{\bx, \by, \bz\}) = \vspan(\{\bx, \by, \bw\}).
$$

## Thought

Translate the problem into something we can read first. By definition, the set $\vspan(\{\bx, \by, \bz\})$ can be read as "every vector that can be written as $c_1\bx + c_2\by + c_3\bz$ with some real numbers $c_1, c_2, c_3$". Similarly, the set $\vspan(\{\bx, \by, \bw\})$ can be read as "every vector that can be written as $d_1\bx + d_2\by + d_3\bw$ with some real numbers $d_1, d_2, d_3$". Therefore, what we need to show is "every vector in the form of $c_1\bx + c_2\by + c_3\bz$ can be writtene as $d_1\bx + d_2\by + d_3\bw$ for some real numbers $d_1, d_2, d_3$" and the other way around.

## Sample answer

Suppose $\bp\in\vspan(\{\bx, \by, \bz\})$. Then $\bp$ can be written as $\bp = c_1\bx + c_2\by + c_3\bz$ for some $c_1, c_2, c_3\in\mathbb{R}$. Since $\bz = \bw - \by$, we know that

$$
\begin{aligned}
\bp &= c_1\bx + c_2\by + c_3(\bw - \by) \\
&= c_1\bx + (c_2 - c_3)\by + c_3\bw,
\end{aligned}
$$

which implies that $\bp$ is a linear combination of $\{\bx, \by, \bw\}$ and $\bp\in\vspan(\{\bx,\by,\bw\})$.

On the other hand, suppose $\bp\in\vspan(\{\bx, \by, \bw\})$. Then $\bp$ can be written as $\bp = d_1\bx + d_2\by + d_3\bw$ for some $d_1, d_2, d_3\in\mathbb{R}$. Since $\bw = \by + \bz$, we know that

$$
\begin{aligned}
\bp &= d_1\bx + d_2\by + d_3(\by + \bz) \\
&= d_1\bx + (d_2 + d_3)\by + d_3\bz,
\end{aligned}
$$

which implies that $\bp$ is a linear combination of $\{\bx, \by, \bz\}$ and $\bp\in\vspan(\{\bx,\by,\bz\})$.

Therefore, we know the two spans are the same.

## Note

The best way to understand a proof is to write it down and explain it to your friends. Take a blank paper, write the proof again without looking at the answer. If you feel like you are missing some part, think about why that part is needed in the proof, and then write it again. Repeat these steps until you can write it smoothly. Once you have done this. Try to explain the proof to your friends.

As an exercise, you may try the same problem with $\bw = 4\bz$.

*This note can be found at Course website > Learning resources.*