Let S1 = {(xy)/ x22y-1), S2 = {(x.y)/ x $3, 15 y 5} and S3 = {(x.y)/ O< x<3, 15ys5}. Then: O s1 and S2 are compact subspaces of R. Only S2 is a compact subspace of R. None of S1, S2 and S2 is compact in R. S2 and S3 are compact subspaces of R.

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 W is a subspace in an inner product space V and V = W + W, prove that V = WW¹.

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Q: Suppose U, W,, W, are subspaces of a vector space V. Show that (Un W,) + (UnW,)Cun (W, + W,) Find…

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Q: 3. Let W = M2x2(C) with zero determinant. Show that W is not a subspace of V = M2x2(C).

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Q: Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

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Q: Verify whether the subsets of W of V = P₂ is a subspace: W = a₂x² + a₁x + ao where a₂ #0

A: The given subset W=a2x2+a1x+a0 where a2≠0 of V=P2. We have to check whether the subset is a supspace…

Q: 6. Let Vi and V2 be subspaces of R" such that Vi C V2. Show that any vector v E R" can be uniquely…

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Q: Let A be an m x n matrix, and let B and C have sizes for which the indicated sums and products are…

A: If the product AB is defined, the entry in row i and column j of AB is the sum of the products of…

Q: Determine, with justification, whether the following sets are subspaces of R4. --| I4 = 0 (a) S (b)…

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Q: If v1 and v2 are in R" and H=Gen{v1, v2}, then H is a subspace of R" Justify

A: Given that : and are in  Let Now, we  have to recall  the definition of subspace of a…

Q: 4. Let W be the subspace of R³ spanned by w the orthogonal projection of 8- onto W. - (2/3) 1/3 and…

A: orthonormal vectors:- A set of non zero vectors v1, v2, v3,......, vn is called orthonormal if…

Q: Please go through each option with explanation

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Q: Let P3(R) denote the R-vector space of real polynomials of degree 2 or less p=a+bX+cX2, where a, b,…

A: Given the polynomial .Let the subspace of  be .The subspace is the set of all polynomials from  such…

Q: If X, is a dense subspace of a normed linear space X show that X X'.

A: We will use basic knowledge of dual of normed linear spaces and the concept of isometric…

Q: (b) For subspaces X and Y, the sum X + Y is a direct sum (denoted X + Y) if every VEX + Y can be…

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Q: 6. Determine if {(x, y): x^2 + y^2 ≥ 8} is a subspace of R^2

A: The given set is {(x, y) | x2 + y2 ≥ 8}. We check whether the set is a subspace of R2.  Since the…

Q: 1. Determine whether w - {[:]•r:a-4-0} X1 E R? : : x1 – 23 = X2 is a subspace of R?.

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Q: Which of the following sets W is NOT a subspace of R3? (Select all that apply.) O w= y : 3x-y+2z-1=0…

A: To identify the which of the given sets are not subspace.

Q: Question 3. Let (X,r) be a topological space and YCX. Show that if the subspace (Y,r,) is connected,…

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Q: Is it possible to solve it using words like claim as my professor did in his example in other photo?…

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Q: Let W be a subset of the vectors in R where W = X2 X4 -x3 = x, - x, }. Show that W is a X3 3. X4…

A: We will use the formal definition of a subspace of a vector space to prove that, W is a subspace of…

Q: 2. Find the projection from R³ onto its subspace spanned by (1, 2, −1)ª and (1,1,2).

A: Given that We  have  to  find  the projection from   onto  its  subspace spanned  by  .Let   be…

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

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Q: Describe all linear maps ƒ : R² → R² that are projections with image the subspace ((1,2)).

A: f: ℝ2→ℝ2Subspace =1, 2

Q: True/False: If V is a subspace of R", then dim(V) + dim(V+) = n. True False

A: This is true and follows from a famous theorem in linear algebra.

Q: Show that V = W₁ © W₂ for the given vector space V and the given subspaces W₁ and W₂. (a) V = R³, W₁…

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Q: Show that the set W={(x,y,1)|x and y are real} is not subspace of R3.

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Q: Let P, be the vector space of polynomials of degree at most 4. (a) Let W1 be the subset of P…

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Q: Describe explicitly a linear transformation from V, (R) into V, (R) which has its range the subspace…

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Q: 2. Determine which of the following is a subspace of R". (a) {(x1,x2,...X) 1 x = 0} (b) {(x1,…

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Q: Be F1(x, x, x) e R³ and F2(a, b,0) e R³ a)show that F1 and F2 are subspaces of R3 b)show that R' =…

A: Solution of the problem as follows

Q: Show that [0,1] and (0,1] as subspaces of R with the usual topology are not homeomorphic.

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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: The set of all points in R3 satisfying x + y - z = 0 is a subspace. Note that the set of point…

A: It is known that any subspace of ℝ3 should contain an additivities identity element 0, 0, 0.

Q: 5) For each of the following subsets of F', determine whether it is a subspace of F³ : (where Fis…

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Q: Let u(g) and v(g) be the Fourier transforms of u(x) and (2) of the product u(z) v(x)? Select one: O…

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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.