Question

Linear Algebra: Can you find an example where \[\mathbb{V}=\mathbb{U}\oplus\mathbb{W}_{1} \] and \[\mathbb{V}=\mathbb{U}\oplus\mathbb{W}_{2}\] (U, W1, W2 are subspaces of V) but \[\mathbb{W}_{1}\neq\mathbb{W}_{2}\]

Answer 1

I'm having trouble with this because we just did a proof were we showed that if the 2 direct sums above were true, them dim W1 = dim W2. But this almost made it seem like W1 = W2, because wouldn't W1 and W2 be orthogonal complements of U?