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Unformatted text preview: M341, Fall 2010, Homework 10. Section 4.4 Problem 10. Since det A =det A T , we may consider the determinant of the matrix with columns u , v , w . These 3 vectors are linearly independent if and only if the three columns of this matrix are pivot columns which happens if and only if there are 3 pivots in the matrix or if and only if the matrix is invertible or if and only if the determinant of the matrix is ̸ = 0. Problem 17. The premise means that in f ( x ) there are two monomials, say a k x k and a ℓ x ℓ , with k ̸ = ℓ and a k ̸ = 0, a ℓ ̸ = 0. Then if we had a dependence relation between f ( x ) and x f ′ ( x ), say u f ( x ) + vx f ′ ( x ) = , we should have u ( a k x k ) + v ( xka k x k − 1 ) = and u ( a ℓ x ℓ ) + v ( xℓa ℓ x ℓ − 1 ) = . The homogeneous system: { ua k + vka k = 0 ua ℓ + vℓa ℓ = 0 should have non trivial solutions in u and v . Since a k ̸ = 0, a ℓ ̸ = 0, this system is equivalent to the system: { u + vk = 0 u + vℓ = 0 the determinant of which is k − ℓ ̸ = 0 and there are no non trivial solutions. Problem 19. a) T = { A v 1 ,...,A v k } is linearly independent ⇒ S = { v 1 ,..., v k } is linearly inde pendent. (otherwise:Premise: T = { A v 1 ,...,A v k } is linearly independent. Conclusion: S = { v 1 ,..., v k } is linearly independent.) Assume T is linearly independent. By contradiction. If S was not linearly independent there would be real numbers a 1 ,...,a k not all zeroes, such that a 1 v 1 + ... + a k v k = . When A acts on each member of this equality, by the properties of the matrix multi plication (distributivity, A ( c v ) = cA ( v ), A = ), we get: a 1 A v 1 + ... + a k A v k = , and this would be a dependence relation in T . But this is not possible since T is assumed to be linearly independent. b) Converse: S is linearly independent ⇒ T is linearly independent. For a counter example don’t try to be smart, just make it simple and take A = O , the zero matrix. (Actually any non invertible matrix would give a counter example.) c) If A is square and non singular (invertible), the converse is true. If A is invertible, A − 1 exists and if we put T = { w 1 ,..., w k } , we have S = { A − 1 w 1 ,...,A − 1 w k } and we may use a) in this new context. Section 4.5 Preliminary. In an ndimensional vector space, for a set S of vectors any 2 of the 3 following properties yield the third one: S # = n ( S contains n vectors)....
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This note was uploaded on 12/05/2011 for the course M 341 taught by Professor Hietmann during the Spring '08 term at University of Texas.
 Spring '08
 HIETMANN

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