5. Consider the vector space P2(R), the set of all polynomial func- tions of degree less than or equal to 2 with real coefficients. Show that the set S = {a.u² + bx + c€ P2(R) | a+b = c} is a subspace of P(R). %3D

Q: 1. Consider the following subspaces of R³: U = {(x,x,y): x,y ER} and W = {(x,-x, 2x): x, y ≤ R} (a)…

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Q: Show that the set W : {(x, y, 5x-4y)| x and y are real numbers} is a subspace of R³.

A: We are give, W=x,y,5x-4y| x and y are real numbers To show: W is a subspace of R3 This can be done…

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

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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: 4.) Determine the dimension of thr subspace of M21 consisting of vectors of the form [a – 2b] a + b…

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Q: Q4(a) Define the null space of an m x n matrix A and prove that it is a subspace of R"

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Q: let S be the collection of vector [x] (3x1 matrix) in R3 that satisfy the given…

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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: 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: Suppose V is an inner product space with dim V = 7 and U is a subspace of V with dimU = 2. Then dim…

A: We know that  The sum of the Dimensions of a subspace and the dimensions of its orthogonal…

Q: Prove that if M is a closed subspace and N is a finite dimensional subspace of a normed space X,…

A: This is a problem of Functional Analysis.

Q: 5. (а) Show that the formula ((: :) (; 3)) = xã + 2yỹ+ zž + 2wũ gives M2x2(R) the structure of an…

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Q: 1. Suppose that A is an nxn non-zero, real matrix and 2 is a fixed real number. Let E ={x=R* : AT =…

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Q: 2. Let P3(R) be the vector space of all polynomials of degree 3 or less (and the zero polynomial).…

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Q: 16. Show that the set V = {(a-2b+c, 3b, c +2a, b-c) | a, b, c = R} is a vector space that is a…

A: as you asked to solve question 16 only:   given: V = {(a-2b+c, 3b, c+2a, b-c) | a, b, c are in R}…

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: (3) Suppose that V is a finite dimensional vector space over R and T € L(V) has no eigenvalues.…

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

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Q: Are the following statements true or false? 1. If W is a subspace of R" and if v is in both W and W,…

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Q: Suppose U and W are two-dimensional subspaces of R3. Show that U∩W≠{0}

A: We want to show that  U∩W≠{0}

Q: 2. Provide an example of a subset U of V that is a subspace of V, as well as an example of subset W…

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Q: If Sz is a subspace of R* of dimension 3, then there cannot exist a subspace S2 of R* such that S, c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠S2 and S2≠R4.

Q: {[:} 5. (a) Consider the subspace W = span and consider the vector -1 1 u = -1 |. Find the vector w…

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

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠ S2 and S2≠R4.

Q: The set {(x,3): x is in R)} is a subspace of R2

A: The given set is x,3: x is in R. It is known that any subspace should contain identity element.

Q: 2. Prove that the set S is a subspace of the vector space V.(5 V = R3x3 and S is the set of upper…

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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: Let A be an mxn matrix. Prove that every vector x in R" can be written in the form x = p+u, where p…

A: Let A be an m×n matrix.Prove: for evert vector x∈ℝn can be written in the form x=p+u, where p is in…

Q: Let P2 denote the vector space of polynomials of degree up to 2. Which of the following subsets of…

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

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Q: Show that the given set V is closed under addition and multiplication by scalars and is therefore a…

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Q: If W is a subspace of R" and ax is in W, where a is a nonzero scalar and x is in R". Then x is in W/…

A: We have to check

Q: 2. The set P3 of all polynomials of degree ≤ 3 with normal polynomial operations is a vector space.…

A: Check which of the following are subspaces of P3.

Q: If W, and W2 are finite - dimensional subspaces of a vector space V, then W + W2 is finite –…

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Q: 3). Determine whether the set of all matrices of the form a b c de f where a +c= 0, and b+d+ f = 0,…

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Q: (2) Find a basis for the subspace W%3D { (а+c, 2a — b, За — 2b — с, b + 2с) | а, b, с € R} of R4.

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Q: Let W1 and W2 be two subspaces of a finite dimensional vector space V over a field F. The formula…

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Q: 7. Consider the subset S of R³, S = {(x, y, z) | 2x – y + 3z = 0}. Show that S is a subspace and…

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Q: 1- Show whether the set of polynomials a, +at +at+at ,where a, =a, +a, is a subspace of P3 or not ?

A: Let V be a vector space over the field F. A non-empty subset S of the vector space V is said to be a…

Q: 5. Let P2 be the vector space of polynomials of degree at most 2. Determine if the vector v = = 10x…

A: Consider the given vector v is in the given subspace. Consider the real numbers a, b such that the…

Q: Let W be an (n – 1)-dimensional subspace of V. Show that V has a basis B satisfying BnW = Ø. |

A: This question is related to vector space . Solution of this question is

Q: Is the set M={1/n | n ε z+} compact as a subspace of R?

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Q: 4. (a) Find a basis for the subspace H of R spanned by the following vectors: 00000 1 (b) What is…

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

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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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Q: (5) Suppose that T E L(V) is such that all subspaces of V of dimension dim(V) – 1 are T-invariant.…

A: Please check step 2 for the solution.!

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

A: Given that  W1  and W2 are the sub spaces of a finite dimensional vector space V.