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Homework 10 - M341 H10(S Zhang 3.2-4 1 Determine whether...

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M341 H10 (S. Zhang) 3.2-4. 1. Determine whether the following sets form subspaces of R 3 . (a) { ( x 1 , x 2 , x 3 ) T | x 1 + x 3 = 1 } (b) { ( x 1 , x 2 , x 3 ) T | x 1 = x 2 = x 3 } (c) { ( x 1 , x 2 , x 3 ) T | x 3 = x 1 + x 2 } (d) { ( x 1 , x 2 , x 3 ) T | x 3 = x 1 , or x 3 = x 2 } ans: (3.2:2) (a) No. (1 , 0 , 0) T belongs to the set, but 2(1 , 0 , 0) T does not. C1 failed. (b) Yes. C ( a, a, a ) T and ( a, a, a ) T +( b, b, b ) T both belong to the set. So both C1 and C2 hold. (c) Yes. Both C ( a, b, a + b ) T ( a, b, a + b ) T + ( c, d, c + d ) T belong to the set. (d) No. (1 , 0 , 1) T and (1 , 1 , 0) T belong to the set, but (1 , 0 , 1) T + (1 , 1 , 0) T = (2 , 1 , 1) T does not. C2 failed. 2. Determine the nullspace 2 1 3 2 , 1 2 - 3 - 1 - 2 - 4 6 3 1 3 - 4 2 - 1 - 1 - 1 3 4 1 1 - 1 2 2 2 - 3 1 - 1 - 1 0 - 5 ans: (3.2:4) The null space of a matrix A is the set of all solutions for Ax = 0. Let us try to solve Ax = 0 by row operations, omitting the right-hand side 0:. (1) 2 1 3 2 1 0 0 1 We get the unique solution 0. So the nullspace N ( A ) = (0 0) is a zero dimensional subspace. (2) 1 2 - 3 - 1 - 2 - 4 6 3 By row operations, 1 2 - 3 0 0 0 0 1 Therefore we can choose x 2 and x 3 as any value. All solu- tions form the nullspace N ( A ) = C 1 2 - 1 0 0 + C 2 3 0 1 0 | C 1 , C 2 R (3) Solving Ax = 0 (omitting 0) 1 3 - 4 2 - 1 - 1 - 1 3 4 By row operations, 1 0 - 1 1 - 1 0 Therefore we can choose any value for x 3 . All solutions form the nullspace N ( A ) = C 1 1 1 1 | C 1 R (4) Solving Ax = 0 (omitting 0). 1 1 - 1 2 2 2 - 3 1 - 1 - 1 0 - 5 By row operations, (we can stop at REF, instead of RREF, 1 1 0 5 1 3 0 Therefore x 2 and x 4 can be free. All solutions form the nullspace N ( A ) = C 1 1 - 1 0 0 + C 2 5 0 3 - 1 | C 1 , C 2 R a=[2 1;3 2];rref(a) a=[1 2 -3 -1;-2 -4 6 3];rref(a) rref([1 3 -4;2 -1 -1;-1 -3 4]) rref([1 1 -1 2;2 2 -3 1;-1 -1 0 -5]) 3. Determine whether the following are subspaces of P 4 . ( P 4 is the set of all polynomials of degree 4 or less) (a) The set of polynomials in P 4 of even degree (b) The set of polynomials of degree 3 (c) The set of all polynomials p ( x ) in P 4 such that p (0) = 0. 1
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(d) The set of all polynomials in P 4 having at least one real root. ans: (3.2:5) (a) No. x + x 2 and x - x 2 are both in the set. But the sum 2 x is not of even degree, and it is not in the set. (b) No. x 3 and - x 3 are both of degree 3. But the sum is not. (c) Yes. If p 1 and p 2 in the set, then p 1 + p 2 and Cp 1 are both in the set as they are polynomials of degree at most 4 and have 0 value at x = 0. (d) No. p 1 ( x ) = x +1 and p 2 ( x ) = x 2 - x are both in the set. But the sum x 1 + 1 has no real root. 4. Determine the following are subspaces of C [ - 1 , 1] (a) { f ( - 1) = f (1) } (b) { f ( - x ) = - f ( x ) } (c) Non decreasing functions f . (d) { f ( - 1) = f (1) = 0 } (e) { f ( - 1) = 0 , or f (1) = 0 } .
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