If V is infinite-dimensional and S-is an infinite-dimensional subspace, must the dimension of V/S be finite? Explain. Prove the correspondenga th

Q: Show that [0, ∞) and (0, 0) ad subspaces of R with the usual topology are not homeomorphic.

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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: Is the union of two subspaces of a vector space, a subspace.

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Q: Find the dimension of the subspace W of R* spanned by set S = {(3,0,1,–2), (–5,4,9,2), (–1,2,5,0)} .

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Q: Prove that every line through the origin in R3 is a subspace of R3.

A: To prove that every line through the origin in R3 is a subspace of R3.

Q: 5. Show that if H and H2 are two orthogonal subspaces of H, then Hin H2 = {0}. Does the converse…

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Q: If y is in a subspace W, then y - ŷ is not zero. Select one: O True False

A: Solve the following

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

A: Given: [0, ∞) and (0, ∞) are subspaces of ℝ. To show: [0, ∞) and (0, ∞)are not homeomorphic.

Q: If X is the subspace of `∞ consisting of all sequences of zeros and ones, what is the induced metric…

A: We have given that ,  X is the subspace of l∞ consisting of all sequences of zeros and ones. We need…

Q: Suppose U and W are subspaces of V for which UUW is a subspace. Show that UCW or WCU.

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Q: Find the orthogonal projection of 3 onto the subspace W of R* spanned by 1 -1 -1 projw (7) = %3|

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Q: Give an example of a subset (not subspace) of R3 that has an infinite number of elements, is closed…

A: A subset W of vector space V over the scalar field K is a subspace of V if and only if following…

Q: Find the orthogonal projection of 4 onto the subspace W of R* spanned by 1 1 -1 projw (7) =

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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: Construct the vector space of all 3-tupies over GF(3). Form a two-dimensional subspace and its dual…

A: Given, the vector space of all 3-tuples over GF(3).

Q: can a subspace of R^n have a dimension less than n.

A: Yes , it can be .

Q: of R". SICSZ, Show that Juppose that s, and Sz are subspaces. With the thet property, dim (s.)s dim…

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Q: Let Y be a closed subspace of a normed linear space X. Show that X is complete X are complete. Y AY…

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Q: Prove that the set of mxn upper triangular matrices is a subspace of Mmxn(F).

A: We have to prove that -

Q: 6. Prove that an open interval (a, b) considered as a subspace of the real line is homeomorphic to…

A: An open interval a,b considered as a subspace of the real line is homeomorphic to the real line.

Q: Let-{ », :x, y, a, b e R and x - a = y – b}.Is W a subspace?

A: We will use the one step test for vector subspace to check if W is a subspace or not. We know that…

Q: If X a normed space and y is a proper proper Linear subspace of X. show that y has empty interior.

A: We proved by contradiction. Suppose Y has a nonempty interior then Y contains a open ball. Then we…

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: Prove that the dual of l° is not isometriç to, 1', but contains a proper subspace that is isometric…

A: Consider that there are bounded functions on l∞ which are not of the form fy=∑xkyk for some x. Let,…

Q: Suppose U1; U2; :::; Um are Önite-dimensional subspaces of V . Prove that U1 + U2 + ::: + Um is…

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Q: How many cyclic subspaces of dimension 9 do the following vector space have? 1. V21(GF(5)) 2.…

A: We’ll answer the first three questions since the exact one wasn’t specified. Please submit a new…

Q: Find the orthogonal projection of onto the subspace W of R' spanned by -1 -1 -1 1 1 projw (7)

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Q: Show that a subspace of a Hausdorff space is Hausdorff.

A: Prove that, a subspace of a Hausdorff space is Hausdorff.

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

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Q: suppose subspace L of R that expand by collection 4 の> find angle between vector 2 and subspace L.

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Q: Provide a situation in where it shows a union of two subspaces of a vector space V and is not…

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Q: In C[−π, π], find the dimension of the subspace spanned by 1, cos 2x, cos2 x.

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Q: Suppose W is a subspace of Rn spanned by n nonzero orthogonal vectors. Explain why W = Rn.

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Q: that set of of degree subspace of P₂ Prove or disprove all Spolynomials is 43 not? Or

A: P2 is vector space of real polynomial of degree less or equal to 2. That is P2=px=a0+a1x+a2x2:…

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

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Q: Verify that if W is a subspace of an inner product space Vand v is in V, then perpw(v) is orthogonal…

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Q: Show that an arbitrary linear combination of two states Vs in the vector space V with the same…

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Q: Give example of subspaces of land l?wich are not closed.

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Q: Show that ifY is a subspace of X, and A is a subset of Y, then the topology a subspace of Y is the…

A: First, we have to show that the topology of A as a subspace of Y is also a subspace of X.

Q: are subspaces of R ?

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Q: Find the dimension of the subspace Hof R2 spanned by

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Q: Prove that span(S) is the intersoction of all subspaces of containing S.

A: Solution: Given that V is a space and S is a subset of V. Let Gs be the intersection of subspaces of…

Q: Let A and B be compact subspaces of a Hausdorff space X. Prove that AUB and AnB are compact.

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Q: Let X be a normed linear space which is non-zero. Prove that X is a Banach space if and only if the…

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Q: The subspace V of R4 is spanned by the vecto (a) Use the Gram-Schmidt process to find a

A: Here, in the question there is given some of the vectors with a subspace spanned by the vectors. We…

Q: Show that the set of vectors of the form a +b forms a subspace of R³. Find a 2а — b basis for this…

A: Given set of vectors of the form: aa+b2a-b Let v1=a1a1+b12a1-b1 and v2=a2a2+b22a2-b2 be two…

Q: If y is not in a subspace W, then y - ŷ is zeo. Select one: True False

A: If y is in subspace W, then the orthogonal projection of y onto W (y') is y itself.  And in that…

Q: A subspace of a completely regular space is completely regular.

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Q: Let Pala, b) be the linear space of polyuomial funetions of degree less than or equal to 2 in the…

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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.