4. Decide if the following statement is True or False: Let U, W, and X be subspaces of a finite dimensional vector space V such that UNW = {0} and UnX = {0}. U +W = U+ X, then W = X. %3D %3D

Q: 3. Show that the set W = {(x, 0) : x e R} is a subspace of R?.

A: In order to show W is a subspace of R2 , we need to prove W is closed under addition. For any two…

Q: If V = R is a vector space and let Hbe a subset of V and is defined as H={(a,b,c): a² +b² = 0,c>0}.…

A: To show that H is not a subspace of a vector

Q: 4) Show that the subspaces of R? are precisely {0}, R2, and all lines in R? through the origin.

A: A subspace of a vector space is itself a vector space that is contained in another vector space.

Q: 4.) Determine the dimension of thr subspace of M21 consisting of vectors of the form [a – 2b] a + b…

A:

Q: Let Y = {(x,y,z)| z=-x}. (a) Is Y a subspace of R? Justify. (b) Is Y a vector space? Justify.

A:

Q: Suppose V is a vector space with a non-trivial subspace W, i.e., W is not {0} and W is not V. If p :…

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

Q: b) Suppose that V is finite dimensional and that T E L(V, W). Prove that there exists a subspace U…

A: In a finite-dimensional Vector space, every subspace is finite-dimensional.

Q: 4. Assume that T E L(V) and that U1 and U2 are T-invariant subspaces of V. Prove that U1 n U2 is…

A:

Q: Let Z E R³ | y Prove that Z is a subspace of R³. = Z

A:

Q: 6. Let U and W be 3-dimensional subspaces of a 4-dimensional vector space V, and suppose that U + W.…

A:

Q: True or False: If U and W are subspaces of a finite dimensional vector space and UnW = {0}, then U°…

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: 10. Let Y = {(x,y,z)| z=-x}. (a) Is Y a subspace of R³? Justify. (b) Is Y a vector space? Justify.

A:

Q: Which of the followings is not a subspace of the vector space V? a) V = Mxn(R),W = {A € VIAT = A} b)…

A:

Q: 5) Determine whether the indicated subset is a subspace of the given vector space. a) (x, y, z), x,…

A:

Q: Determine if the set W of vectors (X1 , X2) in R such that x1² + x2 = 0 is a subspace of R?.

A: We have to determine whether W is a subspace of R^2 or not.

Q: Determine whether W is a subspace of the vector space V.W = {(x, 2x, 3x): x is a real number}, V =…

A: Given: The theorem of test for a subspace explains that if W is a nonempty subset of vector space…

Q: Show that S = {(x, Y, z)|4x – y + 5z = 0, x, y, z E R} is a subspace of R³.

A:

Q: Let S be the subspace of Rn spanned by the vectors x1, x2, . . . , xk. Show that y ∈ S⊥ if and only…

A:

Q: b) Let W = { 0) * x, y e R}. Show that W is not a subspaces of M2x2. %3D

A:

Q: 2. Show that U = {(x,y, x) : , y E R} is a subspace of R$. Then, find a linear complement of U in R.…

A:

Q: 16. Show that the set V = {(a-2b+c, 3b, c +2a, b-c) | a, b, c = R} is a vector space that is a…

A: as you asked to solve question 16 only:   given: V = {(a-2b+c, 3b, c+2a, b-c) | a, b, c are in R}…

Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.…

A:

Q: Suppose V is finite-dimensional and U and W are subspaces of V with W^0 ⊂ U^0. Prove that U ⊂ W.

A:

Q: 3 show that if w is a subspace of fini a dimensional vector space y and dim (W) = dim (V) W = V.

A: This is theorem in vector spaces

Q: Show that the set T= {(a, b, c, d): 2a-3b+Sc-d=0} is a subspace of R'.

A: Given T=abcd∈ℝ4:2a-3b+5c-d=0 Let A be a 1×4 matrix defined by: A=2-35-1 Then the equation…

Q: Which of the following choices is not a subspace of vector space R? a. (x,y,z):z=x-y b.…

A:

Q: Suppose U and W are two-dimensional subspaces of R3. Show that U∩W≠{0}

A: We want to show that  U∩W≠{0}

Q: is

A:

Q: Prove that if S1 and S2 are orthogonal subspacesof Rn, then their intersection consists of only the…

A:

Q: Consider the vectors v1, v2, ..., Um in R". Is span (01,..., īm) necessarily a subspace of R" ?…

A:

Q: If V is a vector space and W is a subset of V that contains the zero vector then W must be a…

A: Given statement If V is a vector space and W is a subset of V that contains the Zero vector and W…

Q: Given vector space V and W, is V a subspace of W?

A:

Q: :If W is a subspace of a finite-dimensional vector space V, then A. dim(W) = dim(V) always B. dim(W)…

A: Given W is a subspace of a finite dimensional vector space V.

Q: Suppose V is finite-dimensional, with dim V = n > 1. Prove that there exist 1-dimensional subspaces…

A: Let us consider that (v1, v2, ..., vn) be a basis for V, and let Ui= span(vi), for i ∈ {1, 2, ...,…

Q: If W is a subspace of ℝ4 whose dimension is 2 then is any set of 2 vector is a basis for W

A: Answer

Q: II. Decide which of the following are subspaces of corresponding vector spaces: C R?

A: Vector space.

Q: W = {(x, y, z): x 2 0},

A: Given the vector space

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: 13. Let U, W be vector subspaces of a vector space V. If dim(U) 3, dim(W) 4, dim(V) = 7 and UnW =…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Consider X = {1, 2, 3, 4, 5} where the topology is {U C X|1 E U}. Find a subspace of X which is…

A: Consider the set X = {1,2,3,4,5} where the topology is {U ≤ X : 1 ∈U}. We have to find a subspace of…

Q: ) Do there exist subspaces S,T, and U of R³ such that S 1 T, SI U, and TiU? if so, provide an…

A: I have considered te subspace generated by orthogonal basis vector.

Q: D. Determine whether the set W = {(a),az) E R|ajaz = 0} is a subspace of the vector space R. %3D the…

A:

Q: Let V and W be subspaces of R", and dim V and dim W are strictly less than n. Assume that no…

A: To prove: If V and W are subspaces of ℝn such that dim V < n, dim W < n, V is not contained in…

Q: Prove that S = {(x, y, z) E R³: 2x + 3y+z = 0} is a subspace of the vector space R3.

A: Explanation of the answer is as follows

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

A:

Q: i) Let V = M2x2(R) and %3D W; = {(: :) ev:a.hceR} and W2 = {(". :) ev :a,be R Prove that W1 and W2…

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.