6. a. Give a basis for the orthogonal complement of the subspace V C R' given by V = {x € R* : xị +x2 – 2x4 = 0, x1 – x2 – x3 + 6x4 = 0, x2+ x3 – 4x4 = 0}.

Q: -4 Find the orthogonal projection ŷ of y = 4 onto the subspace 3 -3 W = Span { uj U2 = 2 Ex: 1.23

A: With the help of definition of projection of a vector on a subspace, we solve this problem.

Q: 1. Let B = be a 1 basis for the subspace V. Find the matrix that projects any 4 vector v E R* onto…

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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: 2. Find the closest point to y in the subspace W spanned by vị and v2. 3 1 -2 y V2 13 3

A: To find  The closest point y in the subspace W spanned by v1 and v2.

Q: For the linear map T = LA : R³ → R³ given by multiplication by A = 300 024 002. find the T-cyclic…

A: Let T be a linear operator on a vector space V and let x be a non zero vector in V. Then the…

Q: Find a basis for the given subspace of R° , and state its dimension for the plane 4 x - 2 y + 5 z =…

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Q: f(x) = 9, g(x) = 7x – 4 and h(x) = -2x². Consider the inner product (p, q) = / p(z)q(x) dæ in the…

A: Let S={v1,v2,v3} represent the orthonormal basis for the subspace. By Gram Schmidt Process,…

Q: -2 Find the orthogonal projection ŷ of y = 3 onto the subspace 2 -2 W = Span { u , U2 Ex: 1.23 : ||

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Q: 3. For a certain subspace W, the orthogonal projection of the vector y = |4 onto W 5 is ŷ = | 0 Use…

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Q: Let W be the subspace of P3 spanned by {t³ +t² – 2t + 1, t² + 1, t³ – 2t, 2t³ + 3t² – 4t + 3}. Find…

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Q: 9. Find the orthogonal projection y of y = onto the subspace 2 0. W Span u = -3

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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: Find a basis for the given subspace of R’, and state its dimension for the plane 4 x - 2 y + 5 z = 0…

A: Solution:-

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: Which of the following is a basis for the subspace {:) X y z x = y+z and x, y,z €RC M2x3(R) ? 0 x 0…

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Q: Let W = {a + bx + cx? + dx³ € Pg : c = a – b+ 2d} be a subspace of P3. Then a basis for W is:

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Q: Let V = P2(t) with inner product defined by (f, g) = So f(t)g(t) dt. Find a basis for the subspace W…

A: An inner product is a generalization of the dot product. A real vector space, an inner product…

Q: Find a basis for the given subspace of R , and state its dimension for the plane 2 x - 4 y+ 5 z = 0.…

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Q: b) Let W = { 0) * x, y e R}. Show that W is not a subspaces of M2x2. %3D

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Q: 8. Let P3(C) be a vector space over C and let D: P3(C) → P3(C) be the differential operator defined…

A: D: P3ℂ→P3ℂ be the differential operatorDpt=dpdtrelative to the basis 1, t, t2, t3

Q: 1 2 Find a basis for subspace V of R4 spanned by the vectors 3 dimension of V? 8 What is the

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Q: 4. Let A be a real m x n matrix. Show that U = {x E R" : Ax = 0} is a subspace of R".

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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: 4) Let X, Y, Z be subspaces of a vector space V and assume that Y CX. Prove that Xn(Y+Z)% =Y+(XnZ).…

A: 4. Given The modular law of sub-space: Let X,Y,Z be subspaces of a vector space V and assume that…

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: 2. Consider the set of matrices W : b : а,b€ R a (a) Show that W is a subspace of M22. (b) Find a…

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Q: Let w = ( { (x.y,z) ER3: x+y-z 0) be a subspace. A basis of w+ is [A].

A: Solution

Q: Let W fa + bx + cx2 + dx'| a+ b 0 and c-3d 0 } be a subspace of Pg. %3D %3D %3D Then the dimension…

A: a+b=0  and c-3d=0 b=-a and c=3d Equation given is; a+ bx+ cx2+ dx3  = a- ax+ 3dx2+ dx3 that can be…

Q: -4 Find the orthogonal projection ŷ of y onto the subspace 4 W = Span { u1 -5 Uj = u2 Ex: 1.23 : ŷ =…

A: Given

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: Prove that the the solution to the equation 2x – y+z= 0 is a subspace of R³.

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Q: Find a basis for the given subspace of R°, and state its dimension for the plane 3 x- 2 y+ 5 z = 0.…

A: We have the vector space R3 , the subspace here is: W={(x,y,z) | 3x-2y+5z=0 ; x, y, z∈R}

Q: 2. Show that {1,x – 1, x (x + 1)} is a basis of the space of polynomials P2 (IR) and that W = {p(x)…

A: If the elements of the set are linearly independent then the set is a basis set. If Wvis a subspace,…

Q: 1 1 onto the subspace W = Span -7 Find the orthogonal projection ŷ of the vector y = u = 2 Ex: 5

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Q: Let W be the subspace of IR' spanned by the vectors 1 4 -2 and -1 -7 Find the matrix A of the…

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Q: | Let W be the subspace of R spanned by the vectors 1 11 -2 and Find the matrix A of the orthogonal…

A: The solution is given as

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: (c) Let M = (xn)el² :x, + x, = 0}. Show that M is a closed subspace and find an orthonormal basis…

A: Please check step 2 for the solution.!

Q: Find the orthogonal projection ŷ of y = -5 onto the subspace 6. -2 W = Span { uį 4 3 u2 %3D -2 Ex:…

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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: 1 1 11. Find a basis for the subspace of R4 spanned by u1 = U2 = U3 and -3 1 1 1 U4 =

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Q: Let A be an mxn matrix. Prove that W = {x ER" | Ax = 0} is a subspace of R", where W is the solution…

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Q: The cosine space F3 contains all combinations y(x) = A cos x+ B cos 2x+C cos 3x. Find a basis for…

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Q: I et f(x) = -3, g(x) = 3x – 9 and h(x) = -9x2. Consider the inner product (P. 9) = [ p(x)q(x) dx in…

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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.!