1. Suppose [tex]V = U \bigoplus W[/tex] where U and W are nonzero subspaces of V. Find all eigenvalues and eigenvectors of [tex]P_{U,W}[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

As long as [tex]\lambda[/tex]=0 is an eigenvalue of [tex]P_{U,W}[/tex], i can prove that [tex]\lambda[/tex]=0 and [tex]\lambda[/tex]=1 are the only eigenvalues and then find the corresponding eigenvectors. can anyone help me to show how [tex]\lambda[/tex]=0 is an eigenvalue of [tex]P_{U,W}[/tex]? does it have to do with the fact that Null([tex]P_{U,W}[/tex])=W?

2. Prove or disprove: there is an inner product on[tex]R^{2}[/tex]such that the associated norm is given by [tex]\|(x_1,x_2)\| = |x_1| + |x_2|[/tex] for all [tex](x_1,x_2)[/tex] in[tex]R^{2}[/tex].

we just started inner product spaces and the wording on this problem confuses me a lot. i can find an example of two vectors that disprove the statement for one inner product induced by the norm, but not for all inner products. the question wants an example that works for all inner products, right? (if i am to disprove it, i mean.) can someone push me in the right direction?

thanks for the help.

edit: nevermind, I think i figured out 1. since W = null P, all nonzero w in W satisfy (P-lambdaI)w=Pw=0, so lambda=0 is an eigenvalue. i still need help with 2 though

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# Homework Help: Two linear algebra questions

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