# hw02 - P x and P y commute then either L x and L y are the...

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STA 6246 Linear Models Fall 2011 Homework #2 Due Friday September 23 1 Let W be the subspace of R 3 spanned by x 1 = (2 , 1 , 0) ± and x 2 = (0 , 1 , 1) ± . Calculate the matrix that gives the orthogonal projection onto W . 2 Let Γ be an orthogonal n × n matrix. Show that Γ preserves lengths and angles, i.e. for any x,y R n , ± Γ x ± = ± x ± and x ) ± y )= x ± y . 3 Let Γ be an n × n matrix. Show that if Γ preserves lengths (i.e. ± Γ x ± = ± x ± for all x ) then Γ is an orthogonal matrix. 4 Suppose x and y are two non-zero vectors in R n , and let the spaces they span be L x and L y , respectively. Let P x and P y be the orthogonal projections onto L x and L y , respectively. Show that if
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Unformatted text preview: P x and P y commute then either L x and L y are the same or they are orthogonal. 5 Let X be an n × p matrix (assume n ≥ p ) and let W be the space spanned by the columns of X . A Show that if rank( X ) = p then X ± X is invertible, and P = X ( X ± X )-1 X ± is the orthogonal projection onto W . B Show that if rank( X ) = r < p and X is any matrix having r columns that form a basis for C ( X ) , then P = X ( X ± X )-1 X ± is the orthogonal projection onto W ....
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