# mat224ps3Soln.pdf - Department of Mathematics University of...

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Department of Mathematics, University of Toronto MAT224H1F - Linear Algebra II Fall 2014 Problem Set 3 Solutions 1. Let T : C 3 C 3 be defined by T ( v ) = Av , where A = - 1 - 1 4 + 4 i 1 - 3 4 + i 0 0 2 + 4 i Find the eigenvalues of T , and find a basis for each eigenspace. Is T diagonalizable? Justify your answer.
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2. Let T : P 2 ( C ) P 2 ( C ) be defined by T ( p ( x )) = 2 xp 0 ( x ) + xp 00 ( x ) . Is T diagonalizable? Justify your answer.
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3. Consider C 3 with the standard inner product (that is, the dot product), and let W = span { (1 , 0 , 1) , ( i, i, i ) } . (a) Find an orthonormal basis for W . (b) Find an orthonormal basis for W . Solution.