(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: et B = · 9, (x − 1)², 3x³) be an ordered basis for P3. Find the coordinate vector of (x) = 3x³ + 2x²…

A: Coordinate vector

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Q: B = (v₁, V₂, (The ia Let V be an inner product space (over C or R), with an orthonormal basis Vn)…

A:

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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…

A:

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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: 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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