Let X be a normed space and Y C X a linear subspace. c) Show that Y is dense in X, if and only if for all a' e X' with r' = 0 on Y, it holds that r' = 0.

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: Prove thatII = {(x, y, z)|x – 2y+ 3z = 0}is a subspace of JR³, %3D -

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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: Using the orthogonal projection of vector Y to get the component of y along the vector space W.

Q: T X = R with standard topology { IneN} CR k= Show that the subspace topology on k is discrete.

A:

Q: Let S = { (x, y, z, w) E R*; x – 2y + w = 0 and 2x + z – w = 0 } . - Prove that S is a subspace by…

A:

Q: (b) If V = R", and U is a subspace of V of dimension m with 0 <m <n, prove that U² = {v € V: B(u, v)…

A: As per our guidelines we are supposed to solve only one problem. For the solution of the first…

Q: Let U be an m-dimensional subspace of Rn andlet V be a k-dimensional subspace of U, where0 &lt; k…

A:

Q: Let W - (a + bx + cx + dx'l a +b-0 and c - 3d = 0) be a subspace of Py. Then the dimension of W is…

A:

Q: Let C[−π, π] be the vector space of functions that are continuous over the interval [−π, π]. Find…

A:

Q: Let W1 and W2 be subspaces of a vector space V having dimensions m and n, respectively, where m > n.…

A: Given, dimension of W1=mdimension of W2=n such that m≥n   We know, W1∩W2 is a subspace of W1W1∩W2 is…

Q: 3) Suppose TE L(V) and U1, . , Um are subspaces of V invariant under T. Prove that U, ...+ Um is…

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Q: #Consider the vector space R' (over R) and ZcR consisting of vectos z=(2,2, 2,), with these…

A:

Q: Suppose Y is a closed subspace of a normed linear space X and xo ¤ Y, x, € X . Prove that there is a…

A: Y is a closed subspace of a normed linear space X and x0∉Y, x0∈X.. We have to prove that there is a…

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: Suppose U1,U;, -..Um are finite-dimensional subspaces of V. Prove that U + Uz + .. +Um is…

A: Suppose U1, U2, ... , Um are finite-dimensional subspaces of V. To prove U1+U2+ ... +Um is finite…

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

A: These 2 are unrelated problems since anyone can be solve without any help of other. By Bartleby…

Q: If V is the vector space of real valued continuous functions , then show that the set W of all d’y…

A: Our Aim is to show that the set W of all solutions of the differential equation3d2ydx2-5dydx+7y=0…

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

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Q: Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V such…

A: Let V be an finite dimensional inner product space and U and W be subspace of V and U∩W=0¯ We have…

Q: et X = {(x,y, z): Vx² + y² + z² <1} be a subspace R° with the standard topology and 0 = (0,0,0).…

A: Given,               X=x, y, z:x2+y2+z2 ≤1  be a subspace of R3 with the standard topology and o=0,…

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

A: Given:- V and W are subspaces of ℝn              dim V&lt;n, dim W&lt;n              Also, there is…

Q: Let V be a vector space over a field F, and let W be a subset of V. Prove that Wis a subspace if and…

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Q: Let V be a vector space over F, let T€ L(V), and let WC be a subspace invariant under T. Prove that…

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

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Q: In the vector space of functions f: R → R, find a basis for the subspace spanned by {sin t, sin 2t ,…

A: Solving question by theorem vector space

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.…

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Q: Show that with the subspace topology Uy on Y, the inclusion map : Y → X is continuous.

A: Given that, UY is subspace topology on Y. Suppose (X, Tx) be topological space and Y is subset of X.…

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: Let W = (a + bx+ cx + dx'|a+ b= 0 and c -3d = 0 ) be a subspace of P Then the dimension of W is…

A:

Q: Find the boundary of the sequence xn in the normalized vector space C [a, b], if it exists. 4 cos/3…

A: Solution: Here the sequence is xnt=4cosn3t5+cosnt and a=-5π,  b=-4π

Q: 12. Let K C R" and let V = { E R" : a1y Vy E K}. Prove that V is a subspace of R".

A: According to our guidelines, we can answer only the first question, the rest can be reposted.

Q: Is T = {(x,y,z)| x+2 y+3z=0} a subspace of R?

A: Subspace

Q: Let W is a finite dimensional subspace of an inner product space V and y is any vector in V. The…

A: Let W be a finite dimensional subspace of an inner product space V. Let T be an orthogonal…

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

A:

Q: 06. Suppose V is an inner product space, and let U,, U, be two subspaces of V satisfying dim (U,) <…

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Q: Let V be a finite dimensional vector space and W CV a subspace. Prove that W has a complimentary…

A:

Q: 3) Explain why S, = {(x, y) E R²:y > 0} and S1 = {(x, x²): x E R} are not subspaces of R?. %3D

A: We will solve both these parts using only basic knowledge of subspaces and vector spaces (in clear…

Q: Is W = ((x,y) | xy=0} a subspace of R??

A:

Q: Let W1 and W2 be subspaces of a vector space V having ilimensions mand n, respectively, where m…

A: Let W1  and W2 be the subspaces of finite dimensional vector space V which satisfy dimW1=m  and…

Q: 3) Let W = { (x, y, z, t) : x = 0, y = -z } be subset of IR*. Is W a subspace of IR* ?

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Q: Suppose that V is a vector space with basis B= {bi | i E I} and S is a subspace of V. Let {B1,... ,…

A: Given that V is a vector space with basis B=bi|i∈I and S is a subspace of V. Let B1,B2,B3,…,Bk be a…

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: Let W = { |xy <0}. Is W a subspace of R?? If so, use generic terms to demonstrate the necessary…

A: W is not a subspace of R2.

Q: Denote by C[0, 1] , the space of all continuous functions defined on the unit interval [0, 1]. Show…

A:

Q: a) Prove that W = {(a,b, 2a + b)} is a subspace of R. (more space on the next %3D if needed)

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Q: 4) Suppose u, v E V, where V is an inner product space, and ||u|| = |||| = 1 and (u, v) = 1. Find u…

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

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Q: Let W = Span{1, t, t2} which is a subspace of the vector space C[−π, π], the set of all continuous…

A:

Q: x=y=z

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.