1. Suppose T: R2 R³ is a linear transformation defined by T([¹]) - - = T Find the matrix of T with respect to the standard bases E2 = {8-0-6} for R2 and R³ respectively. {8.8} an and E3

Q: iv) Find a basis for ker(T). 1o lamsd zied v) Determine the dimension of the ker(T). vi) Find a…

A: Given Transformation convert to the matrix T(w)= 3-6  2-2 4  1 1-2-1-1 2  2 Transform the matrix to…

Q: Let B = {B]. A} mation x→→ Ax. =[%] be a basis for R2 and A = Find the B-matrix for the transfor-

A: Given that The transformation is

Q: 4. (10 points) Let u₁ = , U₂ = - 12₁²-4₁ , V2 = on R2 whose matrix representation with respect to…

A: Given two bases of R2:

Q: 8. Let L : P2 → Pq be the linear transformation defined by L(at? +bt +c) = (a + b)t + (b – c). (a)…

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Q: 1. Let B = {(1,3), (–2, –2)} and B' = {(-12,0), (-4, 4)} be bases for R2, and let A = be the matrix…

A: Matrix is a set of array in which elements are arranged in row and columns. For a matrix and ,…

Q: Consider the bases for M2x2(R) given by a' : and a the stan- dard basis. For the transformation T :…

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Q: Show that a orthogonal 2X 2 matrix corresponds to a rotation in R2 if <det Q = 1 and reflection…

A: To Show: An orthogonal 2×2 matrix corresponds to a rotation in R2 if detQ=1, and reflection in R2 if…

Q: 4. Linear transformation L is given as L(x, y, z) = (2x + z, Y – 2, x+ y). Find LA, the matrix of L…

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Q: For each of the following matrices A: (i) find its rref R; (ii) find the rref R' of AT; (iii) find a…

A: Hello, Since there are multiple questions with multiple subparts. So we'll solve first question…

Q: 3. [32] −1 4-3 4. 5. [. |_3 -4 2 1 2 -1 4,

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Q: +4 M || 3 Find the matrix M of the linear transformation T: R³ → R² given by X1 CH = −3x₁ − x₂ +…

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Q: B ³ = {( ¹₂ ). ( ³ ) } Tv Av where A is a matrix is a basis of R2. T: R2 R2 is a linear…

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: Let 8 = {(1, 3), (-2,-2)} and 8' = {(-12, 0), (-4, 4)} be bases for R² 4 2 A - [83] be the matrix…

A: Note: Since you have posted a question with multiple sub-parts, we will solve the first three…

Q: 15. fi = 1+ 2x – 2x2, f2 = 2 + 3x – 4x2, f3 = 1+4x – x2, B = (f1, f2; f3). 1 T(fi) = 3. 2. ,T(f2) =…

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Q: For this actvity, find the matrix representation [7] for the linear transformation 7 : R³ → R²…

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Q: 2. (2.3 #68) Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B-¹…

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Q: Find the coordinate vector [x]B of the vector x relative to the given basis B. 23) b1 =| b2 and B =…

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Q: Define the linear transformation T : P2 → P2 according to T (p(x)) = p(2x − 5). (a) Compute the…

A: Given T(p(x)) = p(2x − 5)

Q: 2.4 Use this matrix N to rotate the point (0, 3) an angle of 180° anticlockwise around the point (1,…

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Q: -6 V, = and v, = ,and let T: R2→R? be a linear transformation that maps x into x,v, + x,V2. Find a…

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Q: 4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R³, with vectors defined below. %3D…

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Q: Let T : R? → R2×2 be the linear transformation defined by 5x1 -5x2 T x2 2x2 -3x1 - (1)- Ex: 5 : Ex:…

A: Given linear transformation, T:R2→R2×2 is defines by              T x1x2=5x1-5x22x2-3x1

Q: 2 26. a. Find the scaling matrix A that transforms into -1 8 -4

A: (a) Let the scaling matrix A be a00a Consider a00a2-1=8-4(a)(2)+(0)(-1)(0)(2)+(a)(-1)=8-42a-a=8-4 On…

Q: Define T: R² R³ by T ([2]) 211 - 212 - 12 21 +372 Then we can write T(z) – Az where A =

A: If T is a linear transformation from Rn to Rm such that T(x) = Ax then A is an mxn matrix.

Q: 5. Consider V=R². Find the coordinates of = (10, 20) with respect to the ordered basis ß= [(1, 1),…

A: we will find the coordinate vector with respect to the given basis.