Provide a justification for your response to each of the following questions. a. Suppose you have m linearly independent vectors in R". Can you guarantee that they form a basis of R"? b. If A is an invertible m x m matrix, do the columns necessarily form a basis of R"? c. Suppose we have an invertible m x m matrix A, and we perform a sequence of row operations on A to form a matrix B. Can you guarantee that the columns of B form a basis for R"?

Q: Construct a 3 x 3 triangular matrix A so that 1] the vector 2 is in Col A. 3]

A: We know every lower or upper traingular matrix is a triangular matrix.

Q: and Find the matrix A of the orthogonal projection onto W. A =

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Q: Prove that if R is a matrix in echelon form, then a basis for row(R) consists of the nonzero rows of…

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Q: If {e, xe", e2 is a basis of solutions of y" + ay" + by + cy = 0, %3D then a =

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Q: 2. Consider the following matrix and vector 1 2 3 4 A = 2 2 6 8 b = Find the particular solution and…

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Q: Find the basis of the null space of the matrix 1. A = 2 -1 3 6 6 -2 (a) , (b) (c) (d) d. a O b O…

A: Given - A = 12-136-2 To find - Find the basis of the null space of the matrix A.

Q: Consider the following matrices. (To make your job easier, an equivalent echelon form is given for…

A: Given matrix is A=1−32−250−38−2 Given echelon form of matrix A is  A~10−1001−4000 First we will find…

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Q: For each matrix A and ordered basis B, find [LA]8. Also, find an invert- ible matrix Q such that…

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Q: Shown below are matrix A and its rref, R. 3 A = -6 0 b. Find nullity (AT). 050 1 1 5 01 1 2] 00-4 5…

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Q: 1 0 0\ 1 0 0 0 0 0 0 0 0 0 0, Find a basis of P3, where the matrix of D equals No justification is…

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Q: If and B' are bases of a vector space y and -1 P = is the matrix of transition from the base B to…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Suppose aj, az, az, a, and a, are vectors in R, and they are, in order from left to right, the five…

A: According to our guideline we can only answer first three subparts.

Q: In part (a.) wouldn't you have to prove that U = Span(B) as well as B being linearly independent to…

A: Yes, we need to show both the parts. And the question you asked has already been proved in the…

Q: 6. Determine whether the following statements are true or false and provide a justification for your…

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Q: 1 Given the matrix A = 2 3 3 2 3 1 6 5 4 5. 97 85 (a) Find a basis each for N(A) and C(A). What is…

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Q: 4. Find a basis and the dimension of the column space of the matrix A, where (0 1 -2 1 A =| 5 2 1 -3…

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

A: Given  that                                   B=5-1,-13 and                          C=-21,3-2 be…

Q: Consider the following two bases of R: 2 B = 2 1 2 C = 1 1 1 Find the change of basis matrix Pc-B.

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Q: 4. Determine whether the column vectors in each of the following matrices form a basis for R³. If…

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Q: Let B = {v1; v2; v3; v4} be a set of vectors in R, and let A be the matrix made by using the four…

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Q: If A is a n x n matrix with NON zero determinant, then its columns are a basis for a n dimensional…

A: “Since you have asked multiple questions, we will solve the first question for you. If youwant any…

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: What is the standard basis of all 2 by 2 matrix a 9. b d 1 (a) 0. 1 (b) 6. 0. 0. 1 0 0 0 0 (c) 1 0 0…

A: We will find a spanning set which is linearly independent as well. Hence, that will be a basis of…

Q: Let B = {v1, v2, vs} and C = {W1, w2, wy} be two ordered bases for R*. If w = 201 + vy + vy, wg = v…

A: We have to check

Q: 2) Let B and C be bases for R2. If B = {(-1, –1). (2. – 1)} and the change-of-basis matrix from B to…

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Q: In P2, find the change-of-coordinates matrix from the basis B= {1-2t + ?,4 - 7t + 52,5- 8t + 81?} to…

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Q: 2.) Determine whether S = is a basis for R³. If it is write u = (6,3,6) as a linear combination of…

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Q: Only one of the following statements is false (A) If S = {v1, v2, V3, V4, V5} is a linearly…

A: option (c) is a false statement

Q: Prove that if S= {v1, V2,..., Vn} is a basis for V and c is a nonzero constant, then S {cV1; cV2,…

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Q: Consider the following. -6 What is the rank of the matrix with the given vectors as its rows? rank =…

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Q: Let B = {uj = ,U2 = and C = V2 = Find Pg, the transition matrix from the standard basis of R? to the…

A: Given  B =  u1=21 , u2=11  We have to find PB , the transition matrix from the standard basis of ℝ2…

Q: Given the matrix A and the vectors X and b such that: 中 [1-1 15 X2 24 3 13 x=x, and b: b, b. A = 4…

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Q: Let B and C be bases for a vector space V and let the change-of-coordinates 2 -3 matrix from B to C…

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Q: Suppose that A = [c, c2 C3] is a 3 × 3 matrix with columns c;, and suppose that a basis for the…

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Q: Is it true that when an n*n matrix whose columns form a basis of R^n, then A is invertible?

A: We shall answer this in next step

Q: 2. Evaluate: 0020 2020 10 002 0022 Do columns of this matrix form a basis of R4? .

A: Columns of this matrix will form a basis of R4 only if columns forms a linearly independent set…

Q: Given that the matrix A has the reduced row echelon form U: 1 0 3 0 -2 1 0 0 0 0 1 0 0 0 0 2 1 7 2…

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Q: For the matrix below, where C', C², ..., C° denote its columns 1 2 1 3 1 4 3 8 3 15 2 A = 1 2 2 3 11…

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Q: [1 2 4' 2 – 4 0 1 0 – 2 0 1 3 A. (Hint: Need to apply row reduction to AT to identify the lin.…

A: Given                           A=1242-4010-2013 Determine a basic for the row space by choosing…

Q: Given that B = { [1 -2 3], [0 1 0), [0 2 1]} is a basis of R and C = {[-3 -5 6], [2 3 -3], {[- 3 [0…

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Q: (2) Give an example of each of the following: (a) A matrix Q that has orthonormal columns but QQT +…

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Q: In P2 find the change-of-coordinates matrices from basis B to basis C. B= (1+t +t?, 2-t +t?, 2- 2t +…

A: l

Q: In the vector space p, of polynomials of degree at most two, the second column of the matrix of…

A: Given the basis B = e1=1+x ; e2=x+x2 ; e3=2+x2 the basis A = a1=1 ; a2=x ; a3=x2

Q: Find the coordinate matrix of w relative to the orthonormal basis B in R". {(- 5V34 3V34 3V34 5V34 w…

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Q: 10. Suppose that Ja) - En, a«|k), [3) = E bu |k). Show that %3D and that this means that the matrix…

A: We are given two ket vectors |α⟩ and |β⟩ as a linear combination of the individual unit ket vectors…

Q: Given the coordinate matrix of x relative to a (nonstandard) basis B for Rn, find the coordinate…

A: Given the coordinate matrix of x relative to a (nonstandard) basis B for Rn. Where, B=2,-1,0,1 and…

Q: In P2, find the change-of-coordinates matrix from the basis B= {1– 2t + t?,4 - 7t + 5,2 - 2t + 5t?}…

A: See the detailed solution below.

Q: Explain why S is not a basis for M, 2. -{{::}{:}} 0 1 1 0 10 0 1 S is linearly dependent. s does not…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.