(5) Let V be a finite dimensional vector space with basis {v1, ... , Un} and T E L(V) Show the following are equivalent: (a) The matrix for T is upper triangular. (b) T(vk) E Span(v1,.., vk) for all k. (c) Span(v1, ... , Vk) is T-invariant for all k.

Q: (a) Consider the real linear map T : R' - R', expressed as follows: (1 2 -1 49 -1 X1 X1 + 2x2 - x3…

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A: Coordinate vector

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Q: b Ell С pts] Explain how you know that W is a subspace of R¹. er W = a+b+5c = 0, b+2c=0, a +3c+d=0

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Q: Let V be a vector space and suppose that T is a linear operator on V such that T(2u-v)= 3u+5v and…

A: T is a linear operator on V,T2u-v=3u+5vT-2u+2v=-4u-6v

Q: 3) Let S contain vectors of the form f(x) = a + bx + cx² + dx³ : f(-2) = 0 and f'(-4)= 0: Then a…

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Q: 6. Let B be the following ordered basis for R?: в %3 {(1,—1), (2, -3)} . Find the coordinate vector…

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Q: Let w,=(1,0,0,1) and w,=(-1,0,0,1). Let W=(w |w•w, = w-w, = 0}, where w-w. denotes the dot product…

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Q: = 7. Let V be a vector space and suppose B = {u, v} is a basis for V. Prove that B' {ku, v} is also…

A: A subset is called basis if 1) It is an independent set of vectors2) It spans vector  space.

Q: Let fi = 1+2r + 3x², f2 = 2 + 3x + 4x². B = (f1, f2) is a basis for the subspace V = {ao + a1x +…

A: For the given problem on vector spaces we need to determine the coefficient s .

Q: Suppose V is a subspace of R" and suppose {v1, v2, v3} is a basis of V. Decide if the following sets…

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