Let T: P2 → P2 by T(ax? + bx + c) = (12a + 5b – 5c)x² + (-20a – 8b + 10c)x + (10a + 5b – 3c); a the standard basis for P2; ß the basis consisting of x² + 3x, x² – 2, x + 6; v= 4x² + 5x – 2. %3D (a) Find [T]% where a is the standard basis for P,. (b) Find the change of basis matrix from a to ß. (c) Find the change of basis matrix from ß to a. (d) Find [T]. (e) Find [v]g. (1) Find [T(v)]g- (g) Find T(v).

Q: Find bases for the image and the kernel of the matrix Г1 4 A = 2 10 3 16 -1 1] 4 1 7 You should…

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Q: 4. Be T: R3 → RS , Linear copy is represented in the standard bases by the matrix: 1 -1 2 1 2 -2 -3…

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Q: Is B a basis for P2? B = {-x +1 ,2x +3, x² }

A: Given B =  -x+1 , 2x+3 , x2  We know that the standard basis of P2 is   1 , x , x2 .

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Q: Explain why S is not a basis for P,. S = {1, 3x, -4 + x2, 6x} O sis linearly dependent. s does not…

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Q: Explain why S is not a basis for P2. S = {1 - x, 1 - x², -1 – 4x + 5x2} O sis linearly dependent. O…

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Q: Explain why S is not a basis for P2. S = {1, 4x, -9 + x², 7x} S is linearly dependent. O s does not…

A: Given S={1,4x,-9+x2,7x}. We explain why S is not a basis for P2.

Q: Explain why S is not a basis for P2.S = {−3 + 6x, 3x2, 1 − 2x − x2}

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Q: Let, a1, a2 and a3 are the basis for R. [R1] y = |R2 [R3] a1 = |0 a2 = |0 a3 = Find the following:…

A: Since you've asked multiple questions, as per our answering guidelines we are supposed to solve the…

Q: Let B be the basis of R² consisting of the vectors and let C be the basis consisting of Find a…

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Q: The coordinates of p(x)=5+3x+x² relative to the basis {1+x,2-x2,3+x+x²} is O (-1,1,2) none (0,1,1) O…

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Q: Define T:R-→R² by T(x) = Ax. Find a basis B for R² with the property that [T]B is diagonal. - 3 - 3…

A: Given:T:ℝ2→ℝ2 is defined by Tx=AxWhere A=-3-3-1-5To find basis B for ℝ2 such that TB is diagonal.…

Q: Say whether the polynomials P1 (x) = x³–7x²+4x, p2(x) = 3x³-x²+5x-1, p3(x) = x³+4x-x²+1 are linearly…

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Q: The standard basis S = {e1, ez} and two custom bases B = {b1, b2} and C = {c1, c2} for R? are shown…

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Q: If b1, b2 form an orthonormal basis for C and x is any vector in C, then O x4- x bị x b2 0 O x+ x bị…

A: We will find out the required value.

Q: Consider the polynomials p₁ (t)=3-t³, p₂ (t)=2-t², p₂ (t)=t-5t², and p₁ (t)=7-t+3t² −1³. 3 Is {P₁,…

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Q: In the vector space p, of polynomials of degree at most two, the matrix of transition from the basis…

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Q: The set B = {4 – 2x2, 20 + x – 10x2, 28x2 – (55 + 3x)} is a basis for P2. Find the coordinates of…

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Q: 2. Let T : P2 –→ P2 be defined by T(a + bx + cx²) = a – 26 + (2b – 4c)x + 3 cx?. Find a basis C for…

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Q: Consider the following two ordered bases of R³: B = C = a. Find the change of basis matrix from the…

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Q: Explain why S is not a basis for P,. .2 S = (1 - x, 1 - x², -1 – 8x + 9x²} O s is linearly…

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Q: Find a basis for the subspace 96 4a + 6b W = E R2x2 a, b eR} of R2x2- 86 5a Ex: 1 4 6. A basis for W…

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Q: Let P2 denote the vector space of all polynomials in the variable x of degree less than or equal to…

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Q: a) Find the basis for the solution space of I1 +a2 - r3 = 0, -2r1 - 12 + 5r3 = 0, -11 + 4r3 = 0 %3D…

A: (a) given system of linear equation x1+x2-x3=0-2x1-x2+5x3=0-x1+4x3=0 to find - basis for solution…

Q: b) Find the value of k such that (-1,1,1), (1,1,1), (1, -1, k) form a basis of R²

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Q: let B be the standard ordered basis for R2, and let 2 Use Theorem 2.23 and the fact that G )"-(; ) 1…

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Q: Explain why S is not a basis for P2. S = {1 - x, 1 - x², -1 - 4x + 5x2} O sis linearly dependent. O…

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Q: Let P2 denote the vector space of all polynomials in the variable a of degree less than or equal to…

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Q: B = (f1, f2, f3) is a basis for R2[] where fi = 1+ 2x + a², f2 = 2+ 3r + 2x², f3 = 2 + x +3x². If…

A: The correct option is option (e) 1 + 2x²   The detailed solution is as follows below:

Q: The set B = {2 + x², 6 + x + 3x?, 19 + 2x + 9x²} is a basis for P2. Find the coordinates of p(x) (39…

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Q: Explain why S is not a basis for P2. S = {1, 2x, -5 + x², 9x} O sis linearly dependent. O s does not…

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Q: Let B = {x²+x,-2x² + 3x + 3, 2x² – x – 2} be a basis of P2(R)

A: Given,            B=x2+x,  -2x2+3x+3,  2x2-x-2

Q: Let V = P4(R). Find a basis of V containing the polynomials p1(x) = 1 + 2x-x 2+3x 3+2x 4 , p2(x) =…

A: The detailed solution of the problem is given in the next steps.

Q: 1 1 11

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Q: - 3x3 + 7x4 = 0 X1 -7x2 - 4x4 = 0

A: The solution are next step is..

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A: Given that the basis B={1,x+x2,2x2-1} for the vector space P2. Here we have to find the coordinate…

Q: Consider the basis a = {1,1+x, 1+a+x ?} of P 2(R), and suppose 8 is another basis such that the…

A: Please look at step 2 for solution.

Q: V has bases B = (v1, v2, v3) and B' = (w1,w2, W3). If wi vị + 2v2 + 4v3, wz = 2v1 + 5v2 + 6v3, wz =…

A: Given that, B=(v1,v2,v3),B'=(w1,w2,w3) Given that,…

Q: Let V = span {1+ 2x – x² + 3r³, -2+x+x² – x*, -1+ 8r – x² + 7r", 3 + x – 2r² + r'}. Find a basis ß…

A: Step:-1 Note:- To span V we need the polynomials of degree 3, so dimension of V is 4. (think using…

Q: Show that the polynomials P1 = 1+x + x² P2 = x + x² P3 = x2 form a basis for P2.

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Q: Let P2(x) be the vector space of polynomials with real coefficients of degree at most 2 with the or-…

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Q: Consider B find [p(x)]B. {x2 + x, 2 + 2x, 1} an ordered basis of P2(R). If p(x) = 4.x² 2x – 6, %3D

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Q: Suppose A is the matrix for T: R3 - R3 relative to the standard basis. Find the diagonal matrix A'…

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Q: Let P2 be the polynomials of degree no more than 2. Is {x2+x+1,2x+1,3x2+1} a basis for P2 ?

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Q: The set B = {x² – 2, 4x² − (8+3x), 15x² − (28 + 9x)} is a basis for P₂. Find the coordinates of p(x)…

A: Result: Let B be the basis of P2 and let p(x)∈P2. Then the coordinates of p(x) relative to the basis…

Q: ; Let P2 denote the vector space of all polynomials in the variable z of degree less than or equal…

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Q: Let P2(x) be the vector space of polynomials with real coefficients of degree at most 2 with the or-…

A: To find the Transition matrix from B1 to B2 we express vectors of B1 in the linear combination of…

Q: Explain why S is not a basis for P₂. S = {1, 6x, -2 + x²,4x} O S is linearly dependent. OS does not…

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Q: Find the matrix of T(ax + bx + c) = (a - b)x² + 2cx with respect to the basis {x, x, 1} of P2. Use…

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.