4. Consider C" as a real vector space. It has a subspace U := C x {0}"-1. What is the dimension of the quotient space C"/U?

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: 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: Find the orthogonal projection ŷ of the vector y = onto the subspace W = -7 Span u =

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

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Q: Suppose that Tis the projection on W along some subspace W'. Prove that W is T -invariant and that T…

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Q: a) Let V be the set of all polynomials of degree 2. Is Va subspace of P, the vector space of all…

A: If two vectors are in subspace then their sum should also belong to subspace

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…

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Q: 10. Is 2 = {(x, y,z)| x=2 y & z=1} a subspace of R³?

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Q: Let W = {a + bx + cx + dx| a+ b = 0 and c – 3d = 0} be a subspace of P3 Then the dimension of W is…

A: Linear combination of vectors we use to find.

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

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Q: 6. Show that the closure Ỹ of a subspace Y of a normed space X is again a vector subspace.

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Q: a. . Is the set of vectors of the form a subspace of R³? Show your proof. La+2b

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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: Find the orthogonal projection ŷ of y = onto the subspace W = Span { uị u2 5 Ex: 1.23 : ŷ =

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Q: Find the orthogonal projection ŷ of y: 6 onto the subspace W = Span { ui u2 Ex: 1.23 ||

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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: 4. Use the definition of a subspace to prove that the set V:= :, y, z E R, y+ : 0 is a subspace of…

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Q: {{},} Let W be the subspace of R3 spanned by write j = |3 3 as the sum of a vector in W and a vector…

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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: at | f = a is a subspace of RI0,1] if and

A: Two step test

Q: If Sz is a subspace of R" of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: This staement is true.

Q: a Is the set of vectors of the form a subspace of R³? Show your proof. La+2b

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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: a. Is the set of vectors of the form a subspace of R3? Show your proof. La+2b|

A: Since you have asked multiple questions, we will solve the First Question for you. If youwant any…

Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

A: Given : V is any normed vector space and x is any non zero vector and α>0. To find :  A real…

Q: If V is infinite-dimensional and S-is an infinite-dimensional subspace, must the dimension of V/S be…

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Q: 5 1 Find the orthogonal projection ŷ of the vector y onto the subspace W = Span { u = -7 3 Ex: 5 ||

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

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Q: Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1 6. ,

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

A: Vector space.

Q: Q.4 Show that U {(x, -x) |x in R} is a subspace of R?. %3D

A: Consider the given set. U=x,-x|xinR Let, consider that, f,g∈V And f, and g is defined as.…

Q: Let W = {(x, y, 1) | x, y, E R}. Is W a subspace of R3?

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Q: projw(x) is not orthogonal to every vector in the subspace W. O True False

A: We have to check whether the given statement, "projwx is not orthogonal to every vector in the…

Q: 2. Find a basis for the subspace W = {[r y z]T: ax + by + cz = 0, a, b, c e R} of R". What is the…

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Q: snip snip

A: Consider the set S=x,y| xy⩾0. Note that 1, 2,  -2, -1∈S. Because it satisfies 1·2≥0   and   -2·-1≥0.…

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…

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Q: 4. Consider the vector space RR of functions f: R R. Let U = {f: R → R: f(42)=0}. Show that U is a…

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Q: Find the orthogonal projection ŷ of the vector y = onto the subspace W = Span u 2 Ex: 5 C ŷ = %3D

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Q: Is Q = {(x,y,z) | x=2y & z=1} a subspace of R?

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Q: Consider the subspace Wof D, given by W = span (cos x, sin x, x cos x, x sin x). (a) Find the matrix…

A: f(x) = cos x + 2x cos x

Q: For a certain subspace W, the orthogonal projection of the vector y = 5 onto W 4 is ŷ = |1. Use this…

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Q: Find the orthogonal projection ŷ of y = 3 onto the subspace W = Span { u1 = , U2 = 3 Ex: 1.23 %3D

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Q: Let V = R and W = {x|x = (x1, x2, 1)}, Is W a subspace of V?

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Q: Let W = (a + bx + cx² + dx³[ c – 3d = 0 } be a subspace of Py. Then the dimension of W is equal to

A: Here we will find out dimension of W.

Q: Find the projection of the vector v = [1 0 −2]T onto the subspace

A: We have to find the projection of the vector v=10-2 onto the subspace S=Span0-11,011

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

A: