4. Be T: R3 → RS , Linear copy is represented in the standard bases by the matrix: 1 -1 2 1 2 -2 -3 1 -7 0 -2 A =|0 1 1 Find an orthogonal basis for the orthogonal complements of ImT.

Q: 4. Use the 3-space standard basis vectors to find three nonzero vectors a, b, c satisfying axb = a x…

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Q: Let A be a 3 x 3 matrix such that B Find a non-zern vector Null(AT) -5 is a basis of Col(A).

A: Given : Let A be a 3×3 matrix and  B=5-5-2 , -54-2 be the basis of ColA To Find : A non-zero vector…

Q: Write x as the sum of two vectors, one in Span (u1,02,03) and one in Span (u4). Assume that…

A: We have to write x as sum of two vectors, let say x1 and x2 . Where x1 is span of {u1 , u2 , u3} and…

Q: The Gram-Schmidt process changes a basis a1, a2, a3 to an orthonormal basis q1, q2, q3. These are…

A: As per the question we have to explain how the QR decomposition of a matrix A = [a1 a2 a3] gives the…

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: 3 9 A -2 -3 1 9 Find an orthonormal basis of the image of A.

A: Orthonormal basis of a vector space V is a basis of a basis for V whose vectors are orthonormal. Two…

Q: Show that {u,, u2, uz) is an orthogonal basis for R. Then express x as a linear combination of the…

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Q: Given that [x]B -5 is the coordinate matrix of x relative to a (nonstandard) basis…

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Q: Let A € Mnxn (R), & be the standard basis for R", and B = {₁,...,n} be another basis for R". If P is…

A: Given, a matrixA∈Mn×n(ℝ). Given, ε as the standard basis ofℝn and  β={v1,v2,...,vn} as another basis…

Q: Consider the basis B = {(2, 1), (3, 1}} for V = R2 and let {x1, x2} be the dual basis of B. Then for…

A: Given Data: Let us consider the given basis B =2,1,3,1 for V=R2 A=2311 is a matrix of basis is…

Q: If {a, x} is a basis of solutions of x?y" + axy' + by = 0, then a = (А) —2 (В) 2 (C) -3 (D) 3 (E)…

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Q: Find the 2 x 2 matrix P such that Pu is the orthogonal projection of in R² onto the line consisting…

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

A: i just used the definition of matrix of A with respect to basis B'

Q: a) Determine whether or not the set of vectors is a basis of the indicated space: x²-1, x*+x+1,x),P;…

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Q: bị and B := E R³, consider the inner product on R³ defined For a := a2 b2 b3 a3 by (a, B) := (2a1 +…

A: We find the matrix of inner product by finding each possible innner products. The orthonormal basis…

Q: Consider the basis B = {{2, 1), (3, 1}} for V = R2 and let {x1, x2} be the dual basis of B. Then for…

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Q: Consider the basis B = {{2, 1), (3, 1}} for V = R² and let {x1, x2} be the dual basis of B. Then for…

A: Given Data: Let us consider the given basis B =2,1,3,1 for V=R2 A=2311 is a matrix of basis is…

Q: Exhibit a basis for T2, the vector space of all 2 x 2 matrices of real numbers with zero trace.

A: Let V be a vector space over a field F. A set S of vectors in V is said to be a basis of V if  (i) S…

Q: 5. Consider the 4 x 5 matrix A = [u₁|u2|u3|u4| u5], where the columns are U₁= 2 , U2 = 5 Uz = -1 -2…

A: The  4×5  matrix  A= u1 | u2 | u3 | u4 | u5   where  u1=1234 , u2=5102 , u3=   2-1-2   3 , u4=  1…

Q: 2. Apply the Gramm-Schmidt orthogonalization process to the vectors [1, 3, 2] and [1, 0, 1] in order…

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Q: |a,6,d e а a, b, d E R}. S2x2 b d Do three matrices 1 A1 = 0 1 1 0 A2 = A3 1 0 - 1 | form a basis of…

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Q: Find a basis for the orthognonal complement of U = span 1 -2 1 - 1 -2 1 Number of Vectors: 1 Basis…

A: Solution:

Q: Find a basis of Null(A), C(A), and C(AT) for each of the following matrices [1 2] 3 4 (i) A 1 2]…

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Q: Let A 1 3 = 0 3518 5 - 1 1 020 2 1 1 -2 0 8 - -2 3 Find a basis for the row space of A.

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Q: 1 3 1 -1 2 0 5 -2 6 1 A = 3 -1 3 4 1 -3 -2 3

A: We have to find the Nul(A).

Q: 5. Consider the space {) a a + b+c= d : 2a + b = d a + b+ 3c = 0 H = CRª. C d (a) Find a matrix A…

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Q: A basis of R² is given as B₁ = {x₁ = (1,1), x₁ = (1,0) }; Find the change of basis matrix P, from…

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Q: Consider the basis B = {{2, 1), (3, 1}} for V = R² and let {x1, x2} be the dual basis of B. Then for…

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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: A is a 3 x 5 matrix and (v1, v2) is a basis of null(A). The vectors v3, v4, vz E R³ satisfy Avz =…

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Q: 4) Let M₂,2 denote the vector space of 2x2 matrices and W be the set of all 2x2 matrices A= such…

A: Recall  the  definition  of  a  basis  of  a  vector  space :  Let  V be  a  vector  space  over a…

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,…