The sum of two subspaces seems a simple enough concept to me, but I must be misunderstanding it since I don't understand why Axler gives an answer he does in(adsbygoogle = window.adsbygoogle || []).push({}); Linear Algebra Done Right.

SupposeUandWare subspaces of some vector spaceV.

[tex]U = \{(x, 0, 0) \in \textbf{F}^3 : x \in \textbf{F}\} \text{ and } W = \{(y, y, 0) \in \textbf{F}^3 : y \in \textbf{F}\}.[/tex]

The sum is given as follows:

[tex]U + W = \{(x, y, 0) : x, y \in \textbf{F}\}.[/tex]

Whereas it seems to me it should be:

[tex]U + W = \{(x+y, y, 0) : x, y \in \textbf{F}\}.[/tex]

Am I misunderstanding the concept of sum, and it does not really mean that all the elements in [tex]U + W[/tex] should have the form [tex](x, 0, 0) + (y, y, 0)[/tex], or [tex](x+y, y, 0)[/tex]?

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# Vector Space Sums

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