Consider the matrices 1 -2 1 1 -1 1 A -2 4 -2 -2 1 2 -1 -3 1 (a) Find a nonzero vector v E R° which is an element of both of the subspaces Nul (A)C R° and Col(B)C R° (i.e., v E Nul(A) N Col(B)). Explain your reasoning. (b) Are the columns of the product AB linearly independent? Explain why or why not based on your answer to (a), without doing any matrix multiplication.

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Q: The sets B={[1 2 2 1]T;  [1 0 2 0]T;  [2 0 4 -3]T} and B'={[0 2 0 1]T;  [2 1 4 -1]T;  [1 2 2n 4]T}…

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A: on solving this we will get

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Q: Find a basis for the subspace а : a, b, c, d E R with 3a + b=0 and c+ d = 0 C W = 6. 6. d of R4.

A: Since 3a+b=0 Therefore b= -3a  And since  c+d=0  Therefore d= -c

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Q: Let ?1 = (4 3 2 1), ?2 = (−2 0 -1 0), ?3 = (0 3 0 1) a) Find the subspace of R^4 spanned by these…

A: Note: There are many long answer for subparts in the questions. So I will answer the first two…

Q: Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A…

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Q: 30. In the vector space R2x2, 1 A = 0. 1 0 B = C = -1 -3 1 are given. (a) Is {A, B, C} linearly…

A: Linearly independent: A set of vectors v1,v2, ⋯, vn are linearly independent if the following…

Q: Find a basis for the subspace and state the dimension. {[ 2c a − b b − 3c a + 2b ]:a, b, c in R}.

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A: (a)Consider the given matrix, the columns of which spanned V:

Q: Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A…

A: Begin with the matrix as follows to get the reduced row echelon form (RREF):

Q: Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A…

A: We are given,              . So, from the echelon form of A, the pivot columns are column 1, column…

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Q: Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A…

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Q: Consider the subspace W = {(a, b, c) E R³ : a – 26 – 3c = 0} of R³. (b) Is {(5, 1, 1)} a basis for…

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Q: a b c Consider the vector space W = {| d e f: a +e +j = 0} of 3 x 3 matrices having hj 2 trace zero.…

A: Given: The vector space; W=abcdefghj:a+e+j=0 The vector; v=2524-1002-1 To determine: (a.) Two…

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Q: (b) Let A and B be 30 × 29 matrices. Show that there exists a vector c E R30 such that both the…

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Q: Find a basis for the subspace and state the dimension. p– 2q 2р-5r :P,q,r e R а. -2q +2r -Зр+ 6r| 1…

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Q: Find a basis for the subspace W = {(a,b,c) ∈ R3 | b = 2a−3c}.

A: Given below the detailed solution

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Q: Given A be a 2 x 2 matrix with the form listed below: A = [0 a          b 0]  , where V = M2,2.…

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Q: In each part, find a basis for the given subspace of R, and state its dimension. a) All vectors of…

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Q: Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A…

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Q: Find a basis for the subspace 66 E R2x2 a, b E R of R2×2. 7a W 8а + 3b 4b Ex: 1 : 6 A basis for W is…

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Q: Consider the vector space of real 3 x 3 matrices M3,3. i: Consider S {A € M3,3|A = A"} Show that S…

A: Since you have asked multiple questions, we will solve first one for you. If you want other question…

Q: Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A…

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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.