Linear Algebra Done Right Example 1.7

Question

In the book Linear Algebra Done Right I came across this example for the sum of vector spaces.

Did he say the second $W + U$ is still equal to 1.7 because it doesn't matter whether you say $(x+y, y, 0)$ or $(x, y, 0)$ or am I missing something? It seems like this notation can get confusing if that is the case.

Thanks,

Jackson

Answer 1

It does matter whether you say $(x + y, y, 0)$ or $(x, y, 0)$ on the level of individual vectors. But when you consider the collections \begin{align*} \{(x, y, 0)&: x, y \in \Bbb F^3\} \quad \text{and} \\ \{(x + y, y, 0)&:x, y \in \Bbb F^3\}\end{align*}

the individual differences get wiped away; the two sets are equal, as collections of vectors (as others have pointed out).

For a concrete example using $\Bbb F = \Bbb R$, it's easy to see that $(2, 3, 0)$ is in the first set. But it's slightly less easy to see that that we can write $\big(2, 3, 0\big)$ as $\big((-1) + 3, 3, 0\big)$, and thus it belongs in the second set as well. Perhaps vowing to use different variables, i.e., writing the second set as $\{(x' + y', y', 0): x', y' \in \Bbb F^3\}$ will help you ignore the individual differences.

It can be confusing, and it can take some getting used to. Linear algebra is often a transition into higher math classes, so it's good you're paying attention to, and coping with, these details now!

Answer 2

Writing $$\{(x+y,y,0) \in F^3 \mid x,y \in F\}$$ is redundant because the first coordinate takes on all values in $F$ independently of $y$. This is because for any $\eta, y \in F$, we can find an $x$ such that $x+y = \eta$.