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sampleproblems

sampleproblems - ² x y ³ = ² 0 1 1 0 ³² x y ³ 1...

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Sample Problems for Midterm 2 1. Find a basis for ker( A ), where A = 1 2 3 4 0 1 2 3 0 0 0 0 . 2. Let W = x y R 2 | x 0 and y 0 . Is W is a subspace of R 2 ? If yes, prove it. If no, explain why not. 3. Let : R n R m be a linear transformation. Prove that ker( T ) is a subspace of R n . 4. What does it mean for the set of vectors { ~ v 1 , ~ v 2 , ~ v 3 } to be linearly independent? 5. Are the vectors ~ v 1 = 1 2 3 , ~ v 2 = 4 5 6 , ~ v 3 = 7 8 9 linearly independent? You must justify your answer. 6. Let B = { ~ v 1 , ~ v 2 } be a basis for R 2 . Let ~v = 2 ~ v 1 + 3 ~ v 2 . What are the coordinates [ ~v ] B of ~v relative to the basis B ? 7. Let T : R 2 R 2 be a linear transformation given by T x y = 0 1
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Unformatted text preview: ² x y ³ = ² 0 1 1 0 ³² x y ³ . 1 Calculate the matrix of T relative to the the basis B = { v 1 ,v 2 } where ~ v 1 = ± 1 1 ² , ~ v 2 = ± 1-1 ² . 8. Let ~x = 1 1 1 . Find the orthogonal projection of x onto the subspace spanned by { v 1 ,v 2 } , where ~ v 1 = 1 1 , ~ v 2 = 1 . 9. Let W = span { v 1 ,v 2 } , where ~ v 1 = 1 1 , ~ v 2 = 1 1 1 . Find a basis for W ⊥ . 2...
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