Assignment 3

# Assignment 3 - } is a subspace of F n . (b) Now suppose...

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MATH 223, Linear Algebra Winter, 2008 Assignment 3, due in class Wednesday, January 30, 2008 1. Let V = P 2 ( x ) be the vector space of polynomials with real coeﬃcients of degree at most two. For each of the following subsets of V , decide if it is independent, if it is a spanning set for V , and if it is a basis for V . Justify your answers. (a) S 1 = { 1 + x, 1 + x 2 ,x + x 2 } (b) S 2 = { 1 + x - 2 x 2 , 3 - 2 x + x 2 , 5 - 3 x 2 , 13 - 2 x - 5 x 2 } . (c) S 3 = { 1 - x 2 , 2 + x + 3 x 2 , 5 - x + 4 x 2 , 2 x - 7 x 2 } . 2. Find a basis for each of the null space, row space and column space of the following matrix over C . A = 1 2 - i - 3 + 2 i 3 i 0 0 - 1 + i 2 - 2 i - 2 - 3 + i 1 + i 2 - 3 + i - 3 + i - 1 - 3 i 2 i 3 + 6 i - 6 - 10 i - 7 + 6 i 6 + 5 i . Express each row of A as a linear combination of the vectors in your basis for the row space, and express each column of A as a linear combination of the vectors in your basis for the column space. 3. In this problem we suppose that F is a ﬁeld, A is an m × n matrix over F and that W is a subspace of F m . (a) Show that U = { ~v F n : A~v W
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Unformatted text preview: } is a subspace of F n . (b) Now suppose that m = n and A is invertible, and that B = { ~v 1 ,...,~v k } is a basis for W . Show that { A-1 ~v 1 ,...,A-1 ~v k } is a basis for U . 4. An n × n matrix A (over the ﬁeld F ) is called symmetric if and only if A = A T . A is called antisymmetic (or skew-symmetric ) just if A T =-A . Fix n ≥ 2 and let U be the set of all n × n symmetric matrices over F , and W be the set of all n × n antisymmetric matrices over F . (a) Show that each of U and W is a subspace of M n ( F ). For the rest of the problem, V = M n ( F ), and F is either R or C . (b) Show that V = U ⊕ W . What happens if F = Z 2 ? (c) In case n = 3, ﬁnd a basis for each of U and W . (d) Find the dimension of each of U and W for any n . 1...
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## This note was uploaded on 04/29/2008 for the course MATH 223 taught by Professor Loveys during the Fall '07 term at McGill.

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