Question

Is this statement true?
"If U, W are subspaces of \(\mathbb{R}^n\) and dim U + dim W = n, then \(U\cap W = {0}\)"

Answer 1

It should be false. :(
Here is an counter-example.
Suppose a basis of U is \(\{e_1\}\), and a basis of W is \(\{e_1, e_2, ...,e_{n-1}\}\), then dim U = 1, dim W = n-1, dim U +dim W = 1 + (n-1) = n. However, \(U\cap W = \{e_1\}\)

Answer 2

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Answer 3

Thanks :)