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Unformatted text preview: PRACTICE PROBLEMS Here are some practice problems. Their range of difficulty is considerable. They were thrown together fairly quickly; if you think you see a mistake, let us know. You should not read anything significant into the distribution of questions. For example, there are a lot of diagonalizability questions below. That does not mean that there are a lot diagonalizability questions on the exam. It means that we like syllables. Determine if 1 2 3 4 , 2 1 2 1 , 1 2 3 1 , 2 1 2 4 is linearly independent. A linear transformation T : V → W , where V is the set of 3 × 3 matrices and W = R 4 , has a 6dimensional null space. Let A be the standard matrix of T . What is the dimension of (Nul A ) ⊥ ? Let T : R 2 → R 2 rotate vectors clockwise by π , and then reflect them in the line x + y = 0. Find the standard matrix for T . Let r ∈ R n be fixed. Prove that T : R n → R n , given by T : x 7→ proj r x , is a linear transformation. Is it injective? Surjective?...
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 Spring '08
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 Math, Linear Algebra, Matrices, Vector Space

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