Suppose that A is a 7 x 4 matrix and that rank(A) = 3. Fill in the blanks: im(A) is a subspace of and dim(im(A)) = null(A) is a subspace of and dim(null(A)) = row(A) is a subspace of and dim(row(A)) = R7 R³ R 10 R

Q: Which of the following must be true? For any two (3x3) matrices A and B, we must have (A + B)² = A²…

A: Since all the other option is some where incorrect . So the correct option is.....

Q: Determine whether the following are subspaces of R2×2: The set of all 2 × 2 matrices B such that b11…

A:

Q: Determine whether the following are subspaces of R2×2: The set of all singular 2 × 2 matrices

A:

Q: Let AX = B be a nonhomogencous system of linear equations in n unknowns; that is, B+ 0. Show that…

A: Let us denote the solution space by W So, W = { X in Kn : AX = B } which is a subset of Kn

Q: Determine whether the set of all matrices of the form (:: ;). a bc 1o), where a+e 0, and b +d+f 0,…

A: Given: abcdef , a+c=0, b+d+f=0

Q: For each of the following matrices, determine a basis for each of the subspaces R(AT ), N(A), R(A),…

A:

Q: Let A be an m × n matrix, B an n × r matrix, andC = AB. Show that N(B) is a subspace of N(C).

A:

Q: Find a basis for the subspace of P2 spanned by the given vectors.  1 + x, x2, -2 + 2x2, -3x

A: Basis are (1, 1,0), (0, 0,1), (-2, 0,2), (0, -3, 0)

Q: The set of all nxn matrices whose entries are negative or zero is a subspace of Mnn: True False

A: The answer is true

Q: Determine whether the following are subspaces of R2×2: The set of all 2 × 2 lower triangular…

A:

Q: Explain why this statement is true: If V is a subspace of Rn, then there is a n×n matrix A such that…

A: To Proof :-  If V  is a subspace of Rn , then there is a n × n  matrix A such that V = Col A.

Q: For the given subspace, H, (i) find a basis for the subspace, and (ii) state the dimension. 2c (а) Н…

A:

Q: The set of all 3 x 3 matrices of the form 0 0 0 0 0 -5x 0 0 0 where x is a real number, is a…

A: Given :M=00000000-5x

Q: Show that H = : a, b eR is a subspace of M2x2(R). Hint: try to write H as the spa of a set of…

A:

Q: Find the orthogonal complement S⊥ of the subspace S of R3 spanned by the two column vectors of the…

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Q: For an m x n matrix A, the column space is a subspace of Rn. True False

A:

Q: What can be said about Nul A if A is a 5×7 ​matrix? Could it be the zero​ subspace? Why or why​ not?

A:

Q: Solve: (a) If a 7 x 9 matrix A has rank 5, what are the dimensions of the four fundamental…

A: If the order of the matrix is m×n and rank of the matrix is r. Then dimension of column space =r…

Q: In R4 , find the dimension of the subspace spanned by the vectors {(1, 1, 1, 1)T , (1, 2, 2, 2)T ,…

A: Given vectors are {(1, 1, 1, 1)T , (1, 2, 2, 2)T , (1, 1, 2, 1)T , (2, 2, 3, 2)T , (4, 5, 6, 5)T }.

Q: Let B and C be given n x n matrices. (a) Show that U = {X € R"*" : XB = CX} and V = {YB-CY:Y € R"*"}…

A:

Q: Determine whether the following are subspaces of R2×2: The set of all 2 × 2 matrices A such that a12…

A:

Q: Suppose that A is an m × n matrix and that B is an n x p matrix. (a) im(A) related?) Show that…

A:

Q: Let W be the set of all 3 x 3 matrices whose trace is zero. Show that S is a subspace of M33.

A: Let W be the set of all 3×3 matrices whose trace is zero. Show that W is a subspace of M33.

Q: 3. Let A be an arbitrary m x n matrix. Consider all x's, such that Ax = 0, and show that the set of…

A:

Q: d) Let A be an m x m real matrix. If A is skew-symmetric and m is odd, then A is nonsingular. e) The…

A:

Q: The set of all 3 x 3 matrices of the form 1 0 0 1 Lo o 0 0 Select one: True O False + 13x3 where a…

A: Answer: The solution is as given below:

Q: Let A and B denote 3 x 3 matrices and let v = Consider the following subsets of R³. Determine if…

A:

Q: Let B be a fixed mxn matrix and let S be the set of nxp matrices where S={A|BA=0}. Determine if S is…

A:

Q: Suppose a 5x7 matrix A has four pivot columns. What is dim Nul A? Is Col A = R4? Why or why not? dim…

A:

Q: Which of the following subsets of R3x3 are subspaces of Rx3? OA. The diagonal 3 x 3 matrices OB. The…

A:

Q: Let W be the subspace of the space of 3x3 matrices consisting of all matrices A =| def such that A a…

A:

Q: Suppose W is a set of 2 x 2 matrices A with trace A = 0 , and is a subspace of M2,2 Consider the…

A:

Q: а) The subspace H =Span- in R³ has dimension b) If A is a 2 x 2 matrix, and the product A =3: then…

A: a) We will evaluate number of linearly independent vectors in H Let x = 100 y = 400 Clearly,  y =…

Q: Let Mnn be a vector space of n x n matrices. Which of the following is not a subspace of Mnn?

A: The vector space of nxn matrix is Mnn.

Q: Which of the following subsets of R³×³ are subspaces of IR³×3? DA. The 3 x 3 matrices whose entries…

A:

Q: Let A be the matrix consisting of columns equal to the given vectors. A = [w, | w] -1 5 -4 -2 Then…

A:

Q: let W = {x €R3 , 1 such that Ax 2 where A is any 2×3 matrix is a subspace of R³ Select one: True…

A:

Q: If A is an mxn matrix, then the row space of A is a subspace of: O A. Rmn B. R". O C. R". n-m

A: Given that A is an m×n  matrix, then the row space of A is a subspace of :

Q: Which of the following subsets of R³×³ are subspaces of R³×³? A. The 3 x 3 matrices whose entries…

A: The diagonal 3×3 matrices is subspace of R3×3

Q: Find bases for the four fundamental subspaces of the matrix A. 1 4 9 A = 0 8 0

A:

Q: If V = M2x2(R) and W consists of those elements of V in which at least one entry of the matrix is…

A: That's easy. Have a great day!!!

Q: Let A be an m×n matrix. Find all b such that the set of solutions to Ax = b is a subspace and prove.

A: Given: Let A be an m×n matrix.  The objective is to find that the set of solutions to Ax =b is a…

Q: Let A be any 5x7 matrix for which the col(A) has dimension 3, calculate: the nullity(A), and, state…

A:

Q: Let A be a 3 × 2 matrix with rank 2. Give geometric descriptions of R(A) and N(AT ), and describe…

A: Given: Rank of matrix A3x2 is 2. Then the dimension of column matrix A dimR(A)=2 Also given that the…

Q: Consider the subspaces U = span{[1 -4 -5],[-1 –6 -13]} and W = span{[4 -4 4],[-2 0 -2]} of V = R'x3.…

A: Any vector belongs to a space spanned by v1 and v2 is the linear combination of v1 and v2.

Q: Which of the following subsets of RX3 are subspaces of R3x32 A. The 3 x 3 matrices in reduced…

A:

Q: 1. Let A be a 10 x 15 matrix with 6 pivot columns. (i) What is the dimension of Col A? (ii) What is…

A: We are entitled to solve only one question at a time and up to 3 sub-parts only. Given- Let A be a…

Q: If A is an m X n matrix, then N(A) is a subspace of R" Select one: O True False

A:

Q: If A has the same four fundamental subspaces as B, does A = B?

A: Four Fundamental Subspaces: Let be a matrix of order and be its transpose matrix. Then The row…

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.