4. Let V be the vector space of all n x n real matrices, and M EV a fixed matrix. Define W = {A € V | AM = MA}. The set W here is called the centralizer of M in V. Prove that W is a subspace of V.

Q: Let A be a matrixand x be a vector given as follows

A:

Q: Let A be an m × n matrix. Show that every vector v in Rn can be written uniquely as w + u, where w…

A: Given: A is a m×n matrix To show that,  every vector v in Rn be written uniquely in the form, v = w…

Q: Find a basis for the subspace S of R4 consisting of all vectors of the form (a + b, a − b + 2c, b,…

A: We have to find the basis for the subspace S. We have to find the dimension of S.

Q: Prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and, whenever a ∈…

A: We need to prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and,…

Q: Let W, and W2 be subspaces of a vector space V. Prove that V is the direct sum of W, and W2 if and…

A: First assume that V is the direct sum of W1 and W2  W1+W2=V and W1∩W2=0 We have to prove that each…

Q: Q4(a) Define the null space of an m x n matrix A and prove that it is a subspace of R"

A:

Q: Let X be an n-dimensional random vector and the random vector Y be defined as Y = AX + b, where A…

A: It is widely used in statistics . It has applications in almost all multivariate problems .

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: 1. Compute the orthogonal complement of each of the following subspace of the given vecto (a) W =…

A: We have to find orthogonal complement of given subspace: Concept: If x is orthogonal vector to…

Q: If W is a subspace of V and B is a hasis for V, then some subset of B is a basis for W. Select one:…

A:

Q: Prove that if W is a subspace of a vector space V and w1, w2, . . . , wn are in W, then a1w1 + a2w2…

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 A be a fixed m × n matrix. Prove that the set W = {x ∈ Rn: Ax = 0} is a subspace of Rn.

A:

Q: Let W be the subspace of all symmetric matrices in M2,2. What is the dimension of W?

A:

Q: Either prove or disprove that the set V is a subspace V ⊂ R^n defined by  V={x: Ax=λx}where A is…

A:

Q: Suppose that B = {V₁, V2,..., Vk) is an orthonormal basis for a subspace W of R". Let C be the nx k…

A:

Q: Are the following statements true or false? 1. If W is a subspace of R" and if v is in both W and W,…

A:

Q: Let M2×2 be the vector space of all 2X2 matrices, and defined T:M2x2 → M2x2 , such that bi T(A) = A…

A: T:M2×2→M2×2, such thatTA=A+ATA=abcd

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: 2. Provide an example of a subset U of V that is a subspace of V, as well as an example of subset W…

A:

Q: Let V be a finite-dimensional vector space and T be the projection on W along W’, where W and W’are…

A:

Q: {[:} 5. (a) Consider the subspace W = span and consider the vector -1 1 u = -1 |. Find the vector w…

A:

Q: The set {(x,3): x is in R)} is a subspace of R2

A: The given set is x,3: x is in R. It is known that any subspace should contain identity element.

Q: 2. Prove that the set S is a subspace of the vector space V.(5 V = R3x3 and S is the set of upper…

A:

Q: 17. Prove that a subset W of a vector space V is a subspace of V if and only if W + Ø, and, whenever…

A: We Know that Let V be a vector space over the field F and let W E V . Then W will be a subspace of V…

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: Is W a subspace of the vector space? W is the set of all matrices in Mn,n with zero determinants.

A:

Q: Let W be a subspace of R". A set of vectors {v1, ..., Úp} in W is defined to be a basis of W if {v1,…

A: Let ℝn be a vector space. Let W be a subspace of ℝn. Then,  is W={v1→,......,vp→}a basis. Definition…

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: 2. Prove the following given subsets of the vector space, M23, of all 2 x 3 matrices is a subspace.…

A:

Q: Show that the nonempty subset W of a vector space V is a subspace of V if and only if for every pair…

A: Consider a vector space V over a field F. Suppose that W is a non empty subset of V such that…

Q: On the vector space E = Mn(R) of all square matrices, let N(A) = n max |a,l A = (as) € X. Prove that…

A:

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 P be the projection matrix corresponding to a subspace S of Rm. Show that P2 = P

A: Given :Let p be the projection matrix corresponding to a subspace             S of RM

Q: Determine whether the given set S is a subspace of the vector space V. OF. V = R"xn, and S is the…

A:

Q: If W is a subspace of R" and if u is in both W and w then u must be a zero vector.

A: I am going to solve the given problem by using the orthogonal property of a vector.

Q: () If S is a subspace in R that is spanned by then S is spanned by: O a. 1 and 0. 1 1 and 1 c. and…

A:

Q: | Let W be the subspace of R spanned by the vectors 1 11 -2 and Find the matrix A of the orthogonal…

A: The solution is given as

Q: 3). Determine whether the set of all matrices of the form a b c de f where a +c= 0, and b+d+ f = 0,…

A:

Q: Let T be a linear operator on the space V, and let W, ,W hecessary and sufficient condition that…

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: 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…

A:

Q: Determine whether the given subset W is a subspace or not of the given vector space. (a) Let A ∈…

A:

Q: .

A:

Q: Determine whether the given set SS is a subspace of the vector space V.    (2,3,4,5)

A: For subspace test , if u and v is any element of S then au+v must be in S where a is any scalar

Q: 2. Consider the vectors in R', and let W = Span{u, v}. Find a basis for the orthogonal complement…

A: u=213-1 ,v=14-16 To find : Orthogonal complement of the subspace spanned by these  vectors u,v.

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

A:

Q: Let {w1, w2, ..., Wk} be a basis for a subspace W of the vector space V. Let v be a vector in W1.…

A:

Q: 2. Let U and W be a subspaces of an n-dimensional vector space V and suppose further that UNW={0}…

A:

Q: Let W, and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and…

A: Given that  W1  and W2 are the sub spaces of a finite dimensional vector space V.

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.