Let A be a fixed vector in Rn×n and let S be the set of all matrices that commute with A, that is, S = {B | AB = BA} Show that S is a subspace of Rn×n

Q: U is a subspace of Rn, and S is a linearly independent subset of U, then span(S) = U.

A: False

Q: Let W be the set of all 2 × 2 matrices A = " 2 га b] such that Az = 0, where z = . Is W subspace of…

A: A set W of vector space V over field F is said to be subspace of vector space V if W is itself a…

Q: find a basis for the given subspace of R3, and state its dimension. All vectors of the form (a, b,…

A:

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: Let W be the set of all 2 × 2 matrices A = " d. such that Az = 0, where z = |. Is W subspace of M22?…

A: Use the properties of a subspace or definition of subspace. Assume that the condition is true and…

Q: Let A be the vector space of all 2 x 2 matrices over R. Is B a subspace of A? {E ) E A:x + y = z +…

A:

Q: Prove that the set of all n x n matrices A such that AT = -A is a subspace of Mn,n.

A: We have to show that set of all skew symmetric matrix form subspace of Mnn

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: We need to find y.

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

A:

Q: {(: ) : (d) : a, b, cE R. C

A:

Q: Let A be an m × n matrix. Is the set W of all vectors x in Rn such that Ax = 0 a subspace of Rn?…

A: Set W of all vectors X in ℝn such that AX=0 is a subspace. Explain: Let X1 and X2 are two vectors…

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

A:

Q: Let Z be the subspace of all m x n matrices such that the sum of all its entries is zero. Let R be…

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

A: The given subspace is 2ca-bb-3ca+2b. 2ca-bb-3ca+2b=a0101+b0-112+c20-30 Note that, the vectors…

Q: 1. Which of the following sets of matrices form a subspace of M22? (assume a, b range across the…

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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: Let A and B be n × n matrices. (a) Show that AB = O if and only if the column space of B is a…

A: (a) Show that AB = O if and only if the column space of B is a subspace of the null space of A. Here…

Q: Is W a subspace of V? If not, state why. Assume that V has the standard operations. (Select all that…

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Q: Determine which of the following are subspaces of M,n a). The set of all n x n matrices A such that…

A: We have to decide whether the given set form subspace or not.

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: Determine which of the following are subspaces of Mnn- (a) The set of all lower triangular matrices.…

A:

Q: Let RV = M2(R), the vector space of all 2-by-2 matrices with real entries. Determine whether each of…

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

Q: Let W be the set of all n x n diagonal matrices. Is W a subspace of the vector space Mn.n with the…

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Q: (a) If S is the subspace of M6(R) consisting of all upper triangular matrices, then dim S = (b) If S…

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Q: Let A be an mxn matrix. Prove that every vector x in R" can be written in the form x = p+u, where p…

A: Let A be an m×n matrix.Prove: for evert vector x∈ℝn can be written in the form x=p+u, where p is in…

Q: Define the subset W2 of M3(R) to be the set of all matrices A e M3(R) satisfying a1,1 + a2,2 + a3,3…

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Q: Take z in W", and let u represent any element = is in wt) Take z, and z in w", and let u be any…

A: Introduction: There are two vectors x,y∈Rn, if they are orthogonal or perpendicular x·y=0. Notation:…

Q: Let P be the projection matrix corresponding to a subspace S of Rm. Show that (a) P² = P. (b) PT =…

A: This is a problem of linear transformation.

Q: Let M, denote the vector space of all n x n real matrices: Among the following subsets of M, decide…

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

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Q: Let A and B be n x k matrices. satisfying Ax = Bx is a subspace of R. With n = k, show that the set…

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Q: Let S be the subspace of M2(R) consisting of all symmetric matrices (i.e. A=AT). Determine a basis…

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Q: Determine if the set H of all matrices of the form is a subspace of M Here, p, q and 22 r represent…

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Q: Define the subset W1 of M3(R) to be the set of all matrices A e M3(R) satisfying 21,1 + a2,2 + a3,3…

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Q: Let A be the vector space of all 2 x 2 matrices over R. Is B a subspace of A? B = {X € A : X² = X}…

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: Let H be the subset of the vector space M of symmetric matrices of order 3 such that O z+1y 2x H = y…

A:

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

Q: An m×n matrix A is called upper triangular if all entries lying below the diagonal entries are zero,…

A: Proof:

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

A:

Q: Determine whether W is a subspace of V ,  V = Mnn, W is the set of idempotent n X n matrices

A: Let V be a vector space and let W be a nonempty subset of V. Then W is a sub-space of V if and only…

Q: Let V be the vector space of n-square matrices over a ficld K. Show that W is a subspace of V if W…

A: See the attachment.

Q: Which of the following are subspaces of Mnxn(R)? • ) The set of all symmetric matrices A in Mpxn(R).…

A: Since, this is a multi part questions, so I solved first there sub part for you.

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Given vectors are y=466−1, u1=1101, u2=−120−1 and u3=−1011. Now, the formula of orthogonal…

Q: Let V be the vector space of symmetric 2 x 2 matrices and W be the subspace -4 -3 3 W = span{ -3…

A: There are so many vectors does no belong to W but belong to V.

Q: Hi, need help with this calc 3 question, thank you!

A: Here,

Q: Determine which of the following are subspaces of Mpn. (a) The set of all lower triangular matrices.…

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Q: Let A be an mxn matrix. Prove that W = {x ER" | Ax = 0} is a subspace of R", where W is the solution…

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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: Let A and B be m x n matrices. Show that V = {x e R" : Ax = Bx} is a subspace of R".

A:

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.