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18 Nov 2019
9 b and c please?
Critieize the following argument: For any vector v, we have 0v = 0. So the first criterion for subspaces is, in fact, a consequence of the second criterion and could therefore be omitted. Let A be an n times n matrix. Verify that V = (x elementof R^n: Ax = 3x) is a subspace of R^n. Let A and B be m times n matrices. Show that V = (x elementof R^n: Ax = Bx) is a subspace of R^n. a. Let U and V be subspaces of R^n. Define the intersection of U and V to be U union V = (x elementof R^n: x elementof U and x elementof V). Show that U intersection V is a subspace of R^n. Give two examples. b. Is U Union V = (x elementof R^n: x elementof U or x elementof V) always a subspace of R^n? Give a proof or counterexample. Prove that if U and V are subspaces of R^n and W is a subspace of R^n containing all the vectors of U and all the vectors of V (that is, U W and V W), then U + V W. This means that U + V is the smallest subspace containing both U and V. Let v_1, ..., v_k elementof R^n and let v elementof R^n. Prove that Span (v_1, ..., v_k) = Span(v_1, ..., v_k, v) if and only if V elementof Span (v_1, ..., v_k). Determine the intersection of the subspaces P_1 and P_2 in each case: a. P_1 = Span ((1, 0, 1), (2, 1, 2)), P_2 = Span ((1, -1, 0), (1, 3, 2)) b. P_1 = Span ((1, 2, 2), (0, 1, 1)), P_2 = Span ((2, 1, 1), (1, 0, 0)) c. P_1 = Span ((1, 0, -1), (1, 2, 3)), P_2 = [x: x_1-x_2+x_3=0] d. P_1 = Span ((1, 1, 0, 1), (0, 1, 1, 0)), P_2 = Span ((0, 0, 1, 1), (1, 1, 0, 0)) a. P_1 = Span ((1, 0, 1, 2), (0, 1, 0, -1)), P_2 = Span ((1, 1, 2, 1), (1, 1, 0, 1)) Let V R^n be a subspace. Show that V intersection V = [0]. Suppose V and W are orthogonal subspaces of R^n, .ie., v middot w = 0 for every v elementof V and every w elementof W. Prove that V intersection W = [0]. Suppose V and W are orthogonal subspaces of R^n, i, e., v middot w = 0 for every v elementof V and every w elementof W. Prove that V W. Let V R^n be a subspace. Show that V (V). Do you think more is true? Let V and W be subspaces of R^n with the property that V W. Prove that W V. Let A be an m times n matrix. Let V R^n and W R^n be a subspaces. a. Show that {X elementof R^n: Ax elementof W} is a subspace of R^n. b. Show that {y elementof R^n: y = Ax for some x elementof V} is a subspace of R^n. Suppose A is a symmetric n times n matrix. Let V R^n be a subspace with the property that Ax elementof V for every x elementof V. Show that Ay elementof V for all y elementof V. Use Exercises 13 and 14 to prove that for any subspace V R^n, we have V = ((V)). Suppose U and V are subspaces of R^n. Prove that (U + V) = U intersection V .
9 b and c please?
Critieize the following argument: For any vector v, we have 0v = 0. So the first criterion for subspaces is, in fact, a consequence of the second criterion and could therefore be omitted. Let A be an n times n matrix. Verify that V = (x elementof R^n: Ax = 3x) is a subspace of R^n. Let A and B be m times n matrices. Show that V = (x elementof R^n: Ax = Bx) is a subspace of R^n. a. Let U and V be subspaces of R^n. Define the intersection of U and V to be U union V = (x elementof R^n: x elementof U and x elementof V). Show that U intersection V is a subspace of R^n. Give two examples. b. Is U Union V = (x elementof R^n: x elementof U or x elementof V) always a subspace of R^n? Give a proof or counterexample. Prove that if U and V are subspaces of R^n and W is a subspace of R^n containing all the vectors of U and all the vectors of V (that is, U W and V W), then U + V W. This means that U + V is the smallest subspace containing both U and V. Let v_1, ..., v_k elementof R^n and let v elementof R^n. Prove that Span (v_1, ..., v_k) = Span(v_1, ..., v_k, v) if and only if V elementof Span (v_1, ..., v_k). Determine the intersection of the subspaces P_1 and P_2 in each case: a. P_1 = Span ((1, 0, 1), (2, 1, 2)), P_2 = Span ((1, -1, 0), (1, 3, 2)) b. P_1 = Span ((1, 2, 2), (0, 1, 1)), P_2 = Span ((2, 1, 1), (1, 0, 0)) c. P_1 = Span ((1, 0, -1), (1, 2, 3)), P_2 = [x: x_1-x_2+x_3=0] d. P_1 = Span ((1, 1, 0, 1), (0, 1, 1, 0)), P_2 = Span ((0, 0, 1, 1), (1, 1, 0, 0)) a. P_1 = Span ((1, 0, 1, 2), (0, 1, 0, -1)), P_2 = Span ((1, 1, 2, 1), (1, 1, 0, 1)) Let V R^n be a subspace. Show that V intersection V = [0]. Suppose V and W are orthogonal subspaces of R^n, .ie., v middot w = 0 for every v elementof V and every w elementof W. Prove that V intersection W = [0]. Suppose V and W are orthogonal subspaces of R^n, i, e., v middot w = 0 for every v elementof V and every w elementof W. Prove that V W. Let V R^n be a subspace. Show that V (V). Do you think more is true? Let V and W be subspaces of R^n with the property that V W. Prove that W V. Let A be an m times n matrix. Let V R^n and W R^n be a subspaces. a. Show that {X elementof R^n: Ax elementof W} is a subspace of R^n. b. Show that {y elementof R^n: y = Ax for some x elementof V} is a subspace of R^n. Suppose A is a symmetric n times n matrix. Let V R^n be a subspace with the property that Ax elementof V for every x elementof V. Show that Ay elementof V for all y elementof V. Use Exercises 13 and 14 to prove that for any subspace V R^n, we have V = ((V)). Suppose U and V are subspaces of R^n. Prove that (U + V) = U intersection V .
evereadyLv10
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