Consider R 5 with norm ∥(x1, x2, x3, x4, x5)∥ = |x1| + |x2| + |x3| + |x4| + |x5|. The set Y = {((x1, x2, x3, x4, x5) ∈ R 5 : x1 = 0} is a closed subspace of R 5

Q: Let M, N be closed linear subspaces of a Hilbert space H. If M 1 N, prove that M + N is closed.

A:

Q: Let X₁ be a closed subspace and X₂ be a finite dimensional subspace of a normed space X. Prove that…

A: Here we have given that X1 be a closed subspace of a normed space X. So, the quotient space XX1 is…

Q: Let Z be a proper subspace of an n-dimensional vector space X, and let xo € X-Z. Show that there is…

A:

Q: Suppose X is a topological space whose topology is coherent with a family B of subspaces.  Prove…

A: Given X is a topological space, whose topology is coherent with a family B of subspace.Using the…

Q: Let W = {ao + a,r + a2x2|a1, a1, a2 E R, ao + a1 = 0} be a subset of P2, the collection of all…

A:

Q: Suppose V is an interior multiplication space with finite dimension and W1 and W2 are subspaces of…

A: suppose that V is an interior multiplication space with finite dimension and W1 and W2 are subspaces…

Q: A subspace U of R^2 is a subset which satis es three properties: A) 0∈U, B) U is closed under…

A:

Q: (b) Define subset W in R³ as W = {[x1,x2, x3]|4x1 – x2 + 5x3 = 0}. W is a subspace of R3 -

A:

Q: verify that W is a subspace of V. In each case, assume that V has the standard operations. 1. W =…

A: Given: V=ℝ4 and W=x1, x2, x3, 0|x1, x2, x3∈ℝ

Q: Determine whether the following are subspaces of R3 under the usual addition and scalar…

A:

Q: 3. In the following either prove that the set W is a subspace of the vector space V or give a…

A:

Q: Let P2 be the vector space of all polynomials of degree < 2. If W is the set of a polynomials of the…

A: Subspace: Let us consider W⊂V(F), W is said to be subspace of vector space V over field F if it is…

Q: Let S be a subspace of R" and let S be its orthogonal complement. Prove that S- is also a subspace…

A:

Q: Let L = {(x1, x2, x3, X4, X5) E R° | x1 X2 + 2x4 = 0, x2 – 3x4 = 0}. | Prove that L is a subspace of…

A:

Q: . Let L ⊂ R3 be the following subspace: L = {(x1, x2, x3) : x1 + x2 + 3x3 = 0}, and let ℓ be the…

A:

Q: Let T E L(V) and let v1,...Vk E V cigenvectors with corresponding distinct cigenvalues A1, ., Ax of…

A:

Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

Q: Let W={a0+a1x+a2x2|a1,a1,a2∈R,a0+a1=0} be a subset of P2, the collection of all polynomials of…

A: Introduction: A non-empty set of a vector space is referred to a subspace when the set itself…

Q: Let W1 and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and…

A: Consider W1 and W2 be the subspaces of finite dimensional vector spaces V.

Q: Determine the dimension of the subspace of R consisting of all XeR* such that x1 + 2x, = 0 and X3 –…

A:

Q: For each subset W of R² either prove W is a subspace of R² or explain why it is not a subspace. (a)…

A:

Q: Consider the following subset of R": W = {(x1,x2, 83, L4, 25) E R° | r1 = x2 = 13, 14 + 15 = 0}. (a)…

A:

Q: Let X = c, the space of all conver gent sequenes and M == (xn) such that , = 0 then M is not…

A:

Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

A:

Q: 4. Let H be the set of 2 × 2 whose trace is equal to zero. That is [a11 a12 Н- E M2(R) | a11 + a22 =…

A:

Q: 2 Suppose V is finite-dimensional and dim V > 1. Prove that the set of noninvertible operators on V…

A:

Q: Find a basis and the dimension of the subspace W of P3= {ax3+ bx2 + cx + d} defined by W = {PEP3:…

A:

Q: Prove that Co is a Banach space by showing that it is a closed subspace of l

A:

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 W be the subspace of Rª defines as X1 X2 W = {x = ER : : x1 – 3x2 + 4x3 – x4}. %3D x3 X4…

A:

Q: Prove that X is completely regular space if and only if it is homeomorphic to a subspace of a…

A:

Q: Let P, be the set of all polynomial functions in P(R) of the form ·+ anx", for some fixed positive…

A:

Q: Let vi # v2 E R" and define W1 Span{v2}. Use the definition to show W+n W, is a subspace of R".…

A:

Q: Let M be the set of all vectors (or polynomials) x in p (over R) for which x(t)=x(-t) - even…

A:

Q: = E In > 1} be viewed as a subspace of R. What is the subspace topology on X? A. Discrete B.…

A: Let, y=1n: n≥1 be viewed as a subspace of ℝ.

Q: .

A:

Q: Let T be a linear operator on a finite-dimensional vector space V, and suppose that W is a…

A: Let p(x) be the minimal polynomial of T, hence p(x) annihilates T, i.e.,

Q: If X is an inner product space, and A is subspace of X then A = AL %3D O True O False

A: TRUE

Q: Suppose W1 and W2 are subspaces of a vector space, V. Define W as follows: W = {w1 + W2 | W1 E W1…

A: As W1 is a subspace of V , hence (a) 0∈W1 (b) αw1+βw1'∈W1 for every w1 , w1'∈W1 As W2 is a subspace…

Q: Let W1 and W2 be subspaces of a vector space V. Prove that W1 ∪W2 is a subspace of V if and only if…

A: It is given that W1 and W2 are the two subspaces of the vector space V.

Q: 4) Let X, Y,Z be subspaces of V and assume that Y C X. Prove what is known as the modular law of…

A:

Q: so, it. Q2. a) Let P be the set of all polynomials f(x), and let Q be the subset of P consisting of…

A: For Q to be Subspace of P ,we will use one step test for subspace.

Q: Let W be the subspace of P2 defined by all p(x) in P2 such that p(x) = 0. (a) Construct a basis for…

A: (a) A polynomial p(x)=ax2+bx+c satisfies p(1)=0. Hence every p∈W is of the form,…

Q: Let W1,W2,K1,K2, . . . ,Kp,M1,M2, . . . ,Mq be subspaces of a vector space V such that W1 = K1⊕K2⊕·…

A:

Q: 8. If Z is an (n−1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null…

A: Given: n-1 dimensional subspace To find: Scalar multiple

Q: Which of the following is a subspace of the vector space P2 of all polynomials of degree at most 2?…

A:

Q: Let W be a subspace of R". Set w+ = {x € R"|x™w = 0 for each w in W}. Show that W- is a subspace of…

A:

Q: 8. In the following either show that the set W is a subspace of the vector space V or give a…

A:

Q: Suppose V is finite-dimensional. Prove that every linear map on a subspace of V can be extended to a…

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.