Show that if T:V→V is diagonalizable and (Chr(T)) (t) = (₁ – t)... (Ant) for some real numbers A1,..., An then of n det [T] = ₁. ・・ An = II A₁ i=1 for any ordered basis 5 V

Q: 6. (Basis and Dimension.) Say P3(R) = {f: R → R | f(x) = ax³ + bx²+cx+d for some a, b, c, d = R}…

A: The objective of the question is to determine which of the two collections of functions, W₁ and W₂,…

Q: Be S = {v1 = (1,2, 1), v2 = (2,9,0), v3 = (3, 3, 4)} a subset of R and T : R³ → R? T(v1) = (1,0),…

A: Given that  S=v1=1, 2, 1, v2=2, 9, 0, v3=3, 3, 4 be a subset of ℝ3 and     T : ℝ3→ℝ2      Tv1=(1,…

Q: Let C := {1, cos t, cos 2t, . , cos 6t} C V where V is the vector space of all real-valued functions…

A: Given: C:=1, cost, cos2t,…,cos6t⊂V. V=f | f:0,2π→ℝ is the Vector space. J=spanC To prove: C is a…

Q: Let T : R2[x] → R2[x] be given by T(a+bx+cx2 ) = a + b(x+ 1) + b(x + 1)2 . Let v = x + 1. Find a…

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Q: Let T: R* → P2 be defined as T | | = (a + b)t² + (c – d)t + d. a. Find T b. Find a basis for the…

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Q: Let T : R² → R² defined by T() =. Evaluate T(e₁) for the standard basis vector ₁ : T(e₁) = (

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Q: et P E L(C") be defined by, P(z1,..., zn) = (zn, Z1,.….., Zn-1) Determine whether or not P is…

A: Let P∈L(Cn) be defined by   P(z1 , . . . , zn) = (zn , z1, . . . , zn-1)Determine whether P is…

Q: Let f E P(R) be a polynomial of degree n. Let yo,..., Yn be the dual basis of 1, x, x²,..., x". Show…

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Q: 3. The set B = {1-t², t-t²,2-2t + t²} is a basis for P2. Find the coordinate vector of p(t) =…

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Q: { be a basis for R? . Find the vector æ if [æ]R =| 3 Let B 8 x = x = 6. x = x = H 00

A: Given that B=11,21 be the basis for R2. We have to find vector x if xB=23. Since, xB=23 Hence,…

Q: ppose S = &v₁₁ ore that T = nearly independ

A: Answer given below.

Q: Consider the following vectors in V = R³ over F = R. v₁ = (1,0,1) U2 = (0,1,2) √3 = (1,2,0)

A: We have given a set of vectors.To check if the set is linearly independent and form a basis of V(F),

Q: 2. Let B₁ be the basis 2²e², re², e² of V = span(2²e², zeª, z), and B₂ be the basis of that the map…

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Q: 3. Let T : R² → R² be defined by T = X Y -x+y x-y

A: 3. Given that  be defined by:

Q: 4. s] Consider the following vectors in V = R³ over F = R. v₁ = (1,0,1) 2 = (0,1,2) √3 = (1,2,0) (a)…

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Q: Let T be a linear operator on R³ such that the rational form of T is [T] B = 0-20 1 0 for some…

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Q: Consider the ordered basis B = {(1,-1,0), (1, 1, 1), (0, 1, 1)} of R³. the coordinate vector v =…

A: Consider the ordered basis B=1,-1,0, 1,1,1, 0,1,1 of ℝ3.   To Give: The coordinate vector v=-2,-1,0…

Q: Consider the basis S - (V₁, V₂} for R², where v₁ - (1, 1) and v₂ - (1, 0), and let T:R² R² be the…

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Q: Consider the elementary basis vectors & € R¹ (column vectors). Show that the matrix eje +e₂e₂+...+ee…

A: Given that ei→∈Rn are elementary basis.We need to show that e1→e1→T+e2→e2→T+…+ek→ek→T has rank k.We…

Q: T((a, b)) = (a, a + b, a + 2b). Find range T a

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Q: V is a finite dimensional vector space s = {a1,a2, ., an} C V and Sp(S) = Vaccepted 3i fori S is…

A: Note: In the question, ∃ should have been ∀. Otherwise, the statement won't be valid. We are given…

Q: Q No. 2. Show that the set S = {v,,v2,V3,V,}, where v, = (1,0,1,0), v2 = (0,1,-1,2), v, = (0,2, 2,0)…

A: We have to solve

Q: Find the dimension and a basis of U = r2s+t 0 -3r s-t r, s, tER

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Q: Let T : R² → R² by T(x) = A˜, where 1. Find ker T 2. Find a basis for im T. A-[¹₁2] Α = -3 -6

A: 1) Ker T Solution:   1 2     -3 -6   Now, reduce the matrix to reduced row echelon…

Q: Which of the following sets of vectors is a basis for R?? {[:]}. 2 3.

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Q: Consider the basis 5 = {v1, V2, V3} for R°, where v: = (1, 1, 1), v2 = (1, 1, 0), and va = (1,0,0),…

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Q: Define a basis v1, v2 of R² by vị obtained by applying the Gram-Schmidt Procedure to v1, v2? 2 by…

A: Consider the provided question, A Basis of a vector space. Let V be a subspace of Rn for some n. A…

Q: 3. Suppose that {v1, v2, V3} is a basis for V and p:(v;) = dij. Let %3D W1 = 2v1 + v2, W2 = v1 -…

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Q: Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x – 2x?, –1 +x+x², 1…

A: Consider the provided question, Given that,  B=1+x-2x2, -1+x+x2, 1-x+x2B'=1-3x+x2, 1-3x-2x2,…

Q: Consider polynomials with real coefficients and degree at most n. Pn(R)= {ao + a₁x + a₂x²+...+ anxai…

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