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# PPS - PRACTICE PROBLEMS Sketches of solutions Here are some...

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Unformatted text preview: PRACTICE PROBLEMS Sketches of solutions 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. These are independent if the vectors 1 2 3 4 , 2- 1 2 1 , 1 2 3- 1 and 2- 1 2 4 . are independent. The independence of these latter vectors may be determined in the usual way (by reducing the associated 4 × 4 matrix, for example). A linear transformation T : V → W , where V is the set of 3 × 3 matrices and W = R 4 , has a 6-dimensional null space. Let A be the standard matrix of T . What is the dimension of (Nul A ) ⊥ ? This question assumes that you can associate V with R 9 , and thus get a 4 × 9 standard matrix A . Do that; then use the Rank-Nullity Theorem and the fact that (Nul A ) ⊥ = Row A to get the answer. 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 . Take the standard basis vectors, and find how T acts on them; the resulting vectors give the columns of the standard matrix. 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? It is neither — unless n = 1 , in which case it is both. (Instead of messing around with the formula, draw a picture!) Prove that V = ax 2 + bx + c : a + b + c = 0 is a vector space....
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PPS - PRACTICE PROBLEMS Sketches of solutions Here are some...

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