- #1
airpocket
- 4
- 0
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 Linear Algebra Done Right.
Suppose U and W are subspaces of some vector space V.
[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]?
Suppose U and W are subspaces of some vector space V.
[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]?