V. Let T and Q be the matrices -2 -1 -9 -5 -2 0-1 -8 -2 3 -4 T = Q = 2 3. 11 -2 3 -5 -13 -7 3 -1 -4 3 1. Explain why the columns of Q are a basis for R'. 2. Verify that X = span (Q.1.Q.2) and y = span (Q.a. Q.4) are each invariant subspaces under T. 3. Compute the product Q-TQ to determine 4. Judge if T is diagonalizable. If so, find a nonsingular matrix P such that P-TP is diagonal. If not, explain in detail.

Q: 21. Suppose the last column of AB is entirely zero but B itself has no column of zeros. What can you…

A: Introduction: Matrix multiplication, also known as matrix product and the multiplication of two…

Q: Find basis of the column space of the following matrix: A 11 5 -5 -5 3 -7 3 3-9 -4 6 6 0 -9 6 6-9

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Q: Let T,(x1,x2) = (+4x1 – 5z2, +1æ1 – lz2), and T2(x1,X2) = (- 321 – 5æ2 , +5æı + 8x2). Write the…

A: To find the standard matrices for T1,T2 and T10T2.

Q: Let B=(b₁,b₂), C={C₁,C₂), be bases for R² b₁ = (-1,2), b₂ = (1, -1), C₁=(1,2), C₂ = (1,1) Find the…

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Q: A matrix A € Rmxn can also be viewed as a vector, by stacking all its n columns into a single…

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Q: find A*B and B*A for the matrices A= 10 1 2 1 1 3 2 -1 B= 0 0 3 1 2 4 -1 -1 0

A: We have to find AB and BA for the matrices                                             A=10121132-1…

Q: For the matrix M M. [1 -2 -2] 0-5 3 -11 -4 F find three different non-zero vectors that are in the…

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Q: (87) ban 13. Let rr, be vectors in R", and let Q be an m x n matrix. Write the matrix [ Qr₁ Qr, ] as…

A:

Q: .

A: A=02-123-2-1-20 To find: An orthogonal matrix U that diagonalizes .

Q: 6 -1 Evaluate ABT – 3C given that (2 -2 lo -1 2. -2 A = 6 -3 ‚B = and C = -12 7. 3. Use matrix…

A: To solve the given matrix problems.

Q: Let B1 = {T1,t2, t 3} and C, = {w1, w2, w3} be two bases for R³ such that । = - , के = - े;, के; =…

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Q: Exercises A. Consider the matrices below: 07 [1 C = 2 [1 A = 2 -11 2 B = 3 -1 7 5 11 5 3 [12 E = |2…

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Q: Find the change-of-coordinates matrix from B to the standard basis in R². B= 6 -3 9

A: The given basis B=6-3, 19. To find the change of coordinates matrix from B to the standard basis.

Q: Let W be the set of all vectors of the form [3s 4r5t] 2r + 2s - t 3s - r + 5t 3r + 2s + 4t with r, s…

A: The set ( W ) consists of vectors that can be formed by linear combinations of the vectors with…

Q: Make a change of variable, x = Py, that transforms the quadratic form 3x, + 14x, x, + 3x, into a…

A: To find the solution of the given problem we find the matrix representation of the quadratic form.…

Q: Find an orthogonal basis of the column space of the following matrix: 1 -5 6 4 -1 A = -4 -1 -13 9 2…

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Q: In P2, find the change-of-coordinates matrix from the basis B = {1+ 3t², -3 +t-8t²,1 - 4t) to the…

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Q: 5. Consider the vectors u 11. and y= a. Use row operations on an augmented coefficient matrix to…

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Q: 1 3 3 1 3 -2 2 -1 -1 -1 0 -4 1 1 1 -5 A = B = 2 -8 1 2 -3 7 - 3 -1 -2 -8 2 -1 20. Can every vector…

A: As we know basis of the column space of matrix B is formed by columns of B that correspond to the…

Q: V8+h-2 lim OA 1. 12 OB. 192 C. D.

A: Just note that 2 is cube root of 8 and then observe it's derivative of cube root function at x=8.

Q: c. If a matrix has more columns than rows, then the dimension of the column space is bigger than the…

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Q: a. Which of the matrices are orthogonal (has orthonormal columns)? V3 -2 V2 V3 [1 1 -4] -2 1 Find…

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Q: Write x as the sum of two vectors, one in Span ({u₁,42,43) and one in Span (U4). Assume that…

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Q: Find the transpose, conjugate and tranjugate of the following matrices: # = √² = 3₁ 2 AT = 1-1 31 7…

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Q: Determine whether or not the following sets Sof 2 x 2 matrices are linearly independent. Select an…

A: Note: As multiple questions have been asked, so only first question 1 has been solved according to…

Q: Let W be the set of all vectors of the form with r, s and treal. Find a matrix A such that W =…

A: Solution:-

Q: Write v as the sum of two vectors, one in Span (u₁} and one in Span (u2,U3,04). Assume that…

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Q: (a) (BH) By direct multiplication, show that the listed vectors z; are eigenvectors for C1, and find…

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Q: Let T(x) = Ax for the matrix A and [4 -4 -16] -16, b = A = 7 5 Lo 0 1 1 Select true or false for…

A: Given that  A=4-4-1675-16011, b=3-11, c=-451 Select true or false statement.

Q: a. Express each column vector of AB as a linear combination of the column vectors of A. b. Express…

A: a. To express each column vector of AB as a linear combination of the column vectors of A, we can…

Q: Find a basis for the rowspace, columnspace, and nullspace of the following matrices. i 4- i (b) B =…

A: We need to find the basis for the row space of matrix A.  Now, matrix A is: A=222i2i2i21+i1+i-1-i…

Q: b. Let B = {T1, đ2, đ3}, C = {71, 72, T3}, be two orthogonal bases for R³. Show that the…

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Q: In P,, find the change-of-coordinates matrix from the basis B = {1+5t“, - 6 +t- 29t“,1- 6t} to the…

A: Given :   B={1+5t2,-6+t-29t2,1-6t} Standard basis of P2  : C={1,t,t2} Now PC←B =v1,v2,v3 So,…

Q: Consider the bases B1 = {(1,2), (- 3,0)} and B2 = {(3,2), (- 1, –2)} of R. The input to "(2.1)' of…

A: Consider the given vectors. B1=1,2,-3,0B2=3,2,-1,-2 Express the vectors in linear combination.…

Q: Find a basis for the column space of -2 4 4 A = 3 3 4 -2 -16 -8

A: