3 I-0-0-1 Let ui and y = U2 U3 4 (a) Let W = Span{u1, u2, U3}. Write y as the sum of a vector in W and a vector orthogonal to W. Be sure to label which vector corresponds to which subspace. (b) Compute the distance from y to W.

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Using the orthogonal projection of vector Y to get the component of y along the vector space W.

Q: Let W be the subspace spanned by u, and u2. Write y as the sum of a vector in W and a vector…

A: y = 1828u1 = 10-1u2 = 212

Q: Show that the solution set to a system of equations of the form a + a, x = 0 + an x = 0 a m, + ... +…

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Q: Let U and W be subspaces of a vector space V. 1. Prove that UnW is also a subspace of V. 2. Must U…

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Q: Let W be the subspace spanned by u, and uz. Write y as the sum of a vector in W and a vector…

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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: c) Prove that if U and W are subspaces of a vector space V, then their intersection UnW is also a…

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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…

A: We first write given vectors is matrix form and then reduce the given matrix in row echelon form.…

Q: Let W be the subspace spanned by the u's and write y as the sum of a vector in W and a vector…

A: These 2 are unrelated problems since anyone can be solve without any help of other. By Bartleby…

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: If u and v are vectors in R and W is a subspace of R with (u - v) belong to W, then both u and v…

A: If u and v are vectors in R3 and W is a subspace of R3 with (u-v) belong to W, then both u and v…

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

A: Given the set of vectors of the form aba+2b , which can also be expressed as:             W =…

Q: Let U1, U2, Uz be the following subspaces of R: Uj = {(a,b, c) : a = c}, Show that (a) R = U + U2,…

A: Answer

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: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: In this question, the concept of the Orthogonal Decomposition theorem is applied. The vector…

Q: 2 Find the orthogonal projection of u = 1 onto the subspace of R3 spanned by the vectors -3 - 2 V1 =…

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Q: 6. Let U and W be 3-dimensional subspaces of a 4-dimensional vector space V, and suppose that U # W.…

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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: Find the orthogonal projection of the vector v onto the subspace S. v = |2 S =

A: projS(V) = (u1.v/u1.u1)u1 + (u2.v/u2.u2)u2

Q: 6. (a) Determine whether the set of all vectors of the form (a,b,c), where b a+c, is a subspace of…

A: The subspace of R3 is (a,b,c) where b=a+c

Q: 1 1. Let ui = u2 = 2 and u = U3 -5 Let Wi = Span{u, u2} and W2 Span{u3, u4}. %3D (a) Show that the…

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Q: Find all values of h such that vector y will be in the subspace of R spanned by vectors 3. V1, V2,…

A: we will first find the equivalent matrix and then reduce it to row echelon form

Q: Let u, = 2 |, u2 and uz = 0 Note that u, and uz are orthogonal but that uz is not orthogonal to u,…

A: The orthogonal vector is given as follows :

Q: Find the distance from the vector -1 y = -5 10 to the subspace W spanned by -2 1 u 0 and v= 2 ||

A: The given vector is y=-1-510 and the subspace is u=-200, v=12-1.

Q: Let W be the subspace spanned by u, and u,, and write y as the sum of a vector in W and a vector…

A: Given that  Let W be the subspace spanned by u1 and u2​, and To write y as the sum of a vector in W…

Q: 2. Determine if the set S consisting of vectors of the form b where abc = 0 is a subspace of R³.…

A: No, S is not  a subsapce of ℝ3. Explanation: Here,  s1=110 and s2=001 belong to set S. Then,…

Q: 1. Let u and v be vectors in a vector space V, and let H be any subspace of V that contains both u…

A: As per guidelines we have to attempt only 1st question.

Q: 2. Let W be the set of all vectors in R? whose first component is 2. Is W a subspace of R2? Justify…

A: It is not a subspace.

Q: et W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

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Q: 16 In the vector space R, let u= (1, 2, 3), v = (3, 1, 5). w = (3,-4, 7). Prove that the subspace S…

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Q: If u and v are vectors in R and W is a subspace of R with (u + V) belong to W, then both u and v…

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Q: 9. Find the vector closest to y from the subspace of Rª spanned by Vị and v2. 3 1 Vị = 1 , y = V2 =…

A: To find  The vector closest to you from the subspace of R^4 spanned by v1 and v2.

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Let W be subspace spanned by u's. Here , u1 = 1101 , u2 = -131-2 , u3 = -1011 Also , y = 644-1 We…

Q: (a) Determine whether the set of all vectors of the form (a,b,c), where b= a+c, is a subspace of R³.…

A: Consider the subset S of ℝ3 defined as, S=(a,b,c)∈ℝ3 | b=a+c Take  a1,a2,a3,b1,b2,b3…

Q: 2 Indicate if the statement is True or False, and give a brief explanation why. a.A set of 7 vectors…

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Q: Let W be the subspace spanned by u, and u2, and write y as the sum of a vector in W and a vector…

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Q: (b) Find two orthonormal vectors u₁, u₂ that span the same subspace of R³ as V₁ = [3,4,0]¹, V₂ =…

A: Here given vectors v1=340, v2=213

Q: 34. Let W be a subspace spanned by u₁ = (1, 1, 1) and u₂ = (-2, 5, -3). Write vector.orthogonal to…

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Q: () (:) 1 (2) Let u = Describe the set H of the vectors that are orthogonal to u. Is H a subspace?…

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Q: Let u, = u2 = -1. and uz =0| Note that u, and uz are orthogonal but that uz is not orthogonal to…

A: Introduction: If two vectors are perpendicular to one other, they are said to be orthogonal. The dot…

Q: Consider the vectors: V1 = (2,0,0,-2), |V2 = (–14 –7,–12,54). V3 = (0,1 1|-6) V4 =…

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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: 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: In Exercise , find the orthogonal projection of v onto the subspace W spanned by the vectors u?.…

A: v=123 , u1=2-21 and u2=-114

Q: In Exercise , find the orthogonal projection of v onto the subspace W spanned by the vectors u?.…

A: Given :           v = 31-2 ; u1 = 111 ; u2 = 1-10   To find an orthogonal projection of v onto the…

Q: 6. (a) Determine whether the set of all vectors of the form (a,b,c), where b = a+c, is a subspace of…

A: To show the subset W is a subspace , we need to show that (a) 0∈W (b) For every w1 , w2∈W , we have…

Q: Let W be the subspace spanned by u, and u,. Write y as the sum of a vector in W and a vector…

A: There are two vectors u1 and u2 is given. write the vector y as the sum of a vector in W and a…

Q: Find the orthogonal projection of v onto the subspace W spanned by the vectors ui

A: Given:   v = 123 , u1 = 2-21 , u2 = -114    Also, given that the subspace W is spanned by the…