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 = - 1 - 5 2 A basis for R with the property that [T]B is diagonal is { }.

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: 4 Let A = and define T : R? → R? by T(x) = Ax. Find a -2 -1 basis B such that the matrix [T]s is…

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Q: Explain why S is not a basis for P2. S = {1, 4x, -6 + x², 7x} O sis linearly dependent. O s does not…

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Q: Explain why S is not a basis for P. S = {1 – x, 1 – x², -1 – 2x + 3x?} %3D O S is linearly…

A: We will check if the vectors given in set S are linearly independent or dependent also determine if…

Q: Explain why S is not a basis for R². S = {(-6, 2)} OS is linearly dependent. S does not span R². OS…

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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: X, - X 3 X2 - X. X, Let T be the lines transformation so that T= X2 X3 X3 - X2 Find a basis for the…

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Q: Consider the basis S = {v1. V2, V3} for R, where v1 = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1, 0, 0),…

A: Use the basis and the given linear operator to find Tx1,x2,x3 and use it to find T7,14,-1

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: Find an algebraic description of the range of A by row reducing A^T and using the relation between…

A: We know that rank(A) = rank(AT). We need to find an algebraic description of the range of A by row…

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

A: Explanation of the answer is as follows

Q: Find the matrix of T(ax² +bx +c) =(a-b)x² +2cx with respect to the basis {x*,x,1} of P,. Use the…

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Q: Y1 Y2 ER3 [Y3. (x, y) 2x1y1 + X2Y2+5x3Y3, for x = |X3 (a) Find a basis for the orthogonal complement…

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Q: Find a basis fo

A: Given,            A=3-916-26-161-38

Q: Let B be the basis of P3 consisting of the Hermite polynomials 1, 2t, - 2 + 4t“, and - 12t + 8t°;…

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Q: Explain why S is not a basis for P2. S = {1, 7x, -6 + x², 2x} O sis linearly dependent. O s does not…

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Q: find the matna X for T elative to the basis B: T:R'R*, TG j) = (x-J,y-ta), 8'-{(1, -), CO 3)} I D

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Q: Define T:R?→R? by T(x) = Ax. Find a basis B for R with the property that [T]B is diagonal. 2 - 4 A =…

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Q: Find the coordinate vector of p(x) = 3 - x + 4x2 with respect to the basis B = {1, 1 + x, -1 + x2}…

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Q: Let S = {V1, V2} be a basis of R? where v1 and v2 = -3 What element of R? is determined by the…

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Q: This is a textbook question, not a graded question   The first four Hermite polynomials of the…

A: We  have to find the basis for P3

Q: (a) Find the coordinate vector of æ with respect to the ordered basis E = -2 of R3:

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Q: Find a basis for the subspace a W = E R2x2 | 6a + 2b – c = d of R2x2- A basis for W is 1 0 0 Ex: 1 :

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Q: (a) Prove that the set ẞ = {p0(x), p1 (x), ... , Pn(x)} of Lagrange polynomials is linearly…

A: Given: ẞ = {p0(x), p1 (x), ... , Pn(x)} To Prove" (a). Prove that the set ẞ of Lagrange polynomials…

Q: Consider P2(R) with its basis B = (x, x + 1, x² –- 1). Find [x2 + 3x + 5]B.

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Q: Given T : R2 --> R2 defined by T (x, y) = (x - 2y,  x + y) (a) Is T is injecting? (b) Determine…

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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 the orthogonal complement W⊥ of W and give a basis for W ⊥

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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: Explain why S is not a basis for R2. S = {(-6, 6)} O sis linearly dependent. S does not span R². S…

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Q: Let T : P2(R) – P2 (IR) be defined by T(p(x)) = p(x) – 3p'(x), and let ß denote the standard ordered…

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Q: Consider the basis S= {V1, V2, V3} for R, where vV1 (1, 1, 1), v2=(1,1,0), and v3 =(1,0,0), and let…

A: Given:       Tv1=8,-1,16        Tv2=9,0,1         Tv3=-1,17,1      v1=1,1,1       v2=1,1,0…

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: Consider the basis B= {1, x+x2, 2x2-1} for the vector space P,. The coordinate vector of the…

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

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: Let {v1, ... , vn} be an orthogonal basis for Rn and let W = span(v1, ... , vk)· Is it necessarily…

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Q: Let, T: R²³ R³T (x₂√, z) = (x+√+2, 27¹², 24+32) • Test whether the transformation linear or not T is…

A: Given- T: R3→R3: T x,y,z=x+y+z , 2y+z, 2y+3z (a) To test T is linear transformation (b) To find…

Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x², -1 – 6x + 7x2} || O sis linearly dependent.…

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Q: 2. Let B₁ be the basis r²e, ze, e of V = span(r²e², xe, x), and B₂ be the basis is H.6 of w that the…

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Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x², -1 – 6x + 7x²} O sis linearly dependent. O…

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Q: Given basis B = {(1, -1), (2, 3)} and basis C = {(1, 0), (0, 1)}, and linear map T = (x + 2y ; 2x -…

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Q: Explain why S is not a basis for P. S = {1, 2x, =7 + x², 9x} O S is linearly dependent.

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Q: Let V = span {V1, V2}, where V1 and V2 = 1 2 Find a basis S for R3 that includes vi and v2.

A: Let V  = span { v1 , v2}  where   v1= 102 and v2 = 111 Find a basis S for R3 that includes v1 and…

Q: Let {f1, f2, f3} be the dual basis of {e1, €2, €3} for R³, where e₁ = (1, 1, 1), €₂=(1,1,−1), Find…

A: Given basis are e1,e2,e3 of R3 where e1=1,1,1e2=1,1,-1e3=1,-1,-1 Its dual basis are f1,f2,f3 We have…

Q: Define T : P2 → R² via T(a + bx + cx²) = (a + b, b – c) . Find a basis for ker T. Also, find dim(ker…

A: The detailed solution is as follows below:

Q: Consider the basis {1, 1 - t, 1 + t + t2} of the space P2 of polynomials of degree at most two. Find…

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Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x2, -1 – 7x + 8x²} O sis linearly dependent. O…

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Q: d) Consider the linear map T: C→ C whose matrix in the standard basis e1, e2, e3 is (0 1 0V 0 0 1 0…

A: Given linear map T:ℂ3→ℂ3 whose matrix in the standard basis e1,e2,e3 is 010001000. Then, the linear…

Q: Define T:R2→R² by T(x) = Ax. Find a basis B for R² with the property that [T]B is diagonal. 3 A = 1…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.