2. Let S be the subspace of R5 spanned by x = (1, , (a) Find a basis for S- (b) Give a geometrical description of S and S

Q: Find a basis for the subspace of R3 spanned by S = {(2,3, -1), (1,3, -9), (0,1,5)}

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Q: Let 1 1 V1 = V2 1 and let H = span{v1,V2} be a subspace of R* Find an orthonormal basis for H.

A: We will use Gram-Schmidt process to form the orthonormal basis of H. According to uk=vk−∑i=1k−1…

Q: .

A: Here we have to find an orthonormal basis for the subspace spanned by 1, x, x2:

Q: ) Let S be the subspace of Rº given by S = Span Find a basis for S4.

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Q: 4. Find an orthogonal basis of the subspace W of R5 spanned by {ū₁ = (1, 1, 1, 0, 1), 2 = (1, 0, 0,…

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Q: Find a basis for the subspace of R3 spanned by s. S = {(1, 5, 9), (-1, 6, 9), (2, 6, 1)}

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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: Let {x1, x2, . . . , xk, xk+1, . . . , xn} be an orthonormal basis for an inner product space V. Let…

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Q: 4 Let S be the subspace of R* spanned by x1 = (1,0, – 2, 1)" and x2 Find a basis for S. (0, 1, 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 Q5. Find an orthonormal basis of the subspace spanned by the vectors

A: The solution is given as

Q: Find a basis for the subspace of R3 spanned by S.S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)}

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Q: 7 4. 3 and let W the subspace of IR spanned by u and v. Find a basis of W, the orthogonal complement…

A: The objective is to find orthogonal complement of W

Q: Let S=v1,v2,v3,v4, where v1= (1,1,2,1), v2= (1,0,−3,1),v3= (0,1,1,2),v4= (0,0,1,1), and v5=…

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Q: Find the orthogonal projection of 4 onto the subspace W of IR' spanned by 1 1 -1 1 1 1 1 projw (7)…

A: I am solving this in step 2

Q: 4. Let S be the subspace of R4 spanned by x = (1,0,-2,1)7 and x2 = (0,1,3, -2)T. Find a basis for S

A: Consider a matrix A as follows.

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: Let Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R³ spanned by…

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Q: Let 1 -3 and let W the subspace of R' spanned by u and v. Find a basis of W-, the orthogonal…

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Q: Let -4 Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R4 spanned…

A: Given, x→=22-40,       y→=-5-101,           z→=-57-2-3 Gram-Schmidt method: Let {x,y,z} be any set…

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: 4. Let S be the subspace of P₂ defined by S = span{1-x, 1+x, 5, 1-x², 2x}. Find dim (S), and show…

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Q: Find the orthogonal projection of 5 16 onto the subspace W of R spanned by 4 -4 and 24 projw (7)

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Q: Find the orthogonal projection of -5 -1 16 onto the subspace W of IR³ spanned by 4 -4 and 2 24 projw…

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Q: Find a basis for the subspace of R³ spanned by S. S = {(1, 4, 7), (-1, 5, 7), (2, 5, 1)}

A: The matrix form the vectors is: A=1-12455771

Q: Suppose {x,y} is a basis for a subspace W of Rn. Let u = x + y and v = x − y. Show that {u,v} is…

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Q: Find a basis for the subspace of R3 spanned by S. S = {(5, 9, 9), (1, 2, 2), (1, 1, 1)}

A: To find the basis for the subspace of R3 spanned by the given set of vectors S.

Q: 26 Let V be the vector space of all 2 x 2 matrices and let W be the set of matrices of the form X -x…

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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: 9. Find a basis for the orthogonal complement of the subspace W=Span

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Q: Find a basis for the subspace of R3 spanned by S. S = {(1, 3, 6), (-1, 4, 6), (2, 4, 1)} EEE

A: Basis of subspace of ℝ3 i) It must be linearly independent. ii) Its span ℝ3

Q: 2, Find bases for the fundamental subspaces; R(A), R(A"), N(A), and N(A"), given the matrix 1 -2 2 A…

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Q: 1. Is the following set of vectors in R³ linearly dependent: {(1,0, 3), (2, 1, –2), (0, – 1, 8),…

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Q: 4) Find a basis for the subspace of R' spanned by the vectors. {(1,1,0,0), (0,0,1,1), (-2,0,2,2),…

A: We have fond the maximal linearly independent subset of the given set.

Q: 5. Let S = {x + 1,x2 – 2, x – 1,3}. Find a basis for the subspace W that is spanned by S.

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Q: 4. Let S be the subspace of P2 defined by S = span{1-x, 1+x, 5, 1-x², 2x}. Find dim(S), and show how…

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Q: Find the orthogonal projection of -13] 15 onto the subspace W of R4 spanned by 4 and 5 -2 -2…

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Q: 2 2 , find the closest point to v in the subspace W spanned by 6 and 3 Given i = 6. 1 12

A: To find  The closest point to V in the subspace spaneed by the given vectors.

Q: Find a basis for the subspace W of R³ orthogonal to {(1,1,3, 4, 1), (1, 2, 1, 2, 1)}.

A: We Use Gram Schmidt Process to Find Orthogonal basis. Use Standard Inner product as <(u1, u2),…

Q: Find the orthogonal projection of 10 onto the subspace W of R° spanned by 9. and 3 -1-10 3 projw (i)…

A: To find  The orthogonal projection of v onto the subspace of W spanned by the given vectors.

Q: 1. Is the following set of vectors in R³ linearly dependent: {(1,0,3), (2, 1, –2), (0, – 1,8), (7,2,…

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Q: 5. Find basis for the subspace u- of R for the vector u = orthogonal subspace of u) (1,3,-4). (Here…

A: As we know that S=v1,v2,...vk be a set of vectors in Rn  then S is called an Orthogonal if and only…

Q: [2] --31 Let W be the subspace spanned by 2 5 Find an orthonormal basis for W.

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Q: 6. Let S be the subspace of R3 that consists of all solutions to the equation x+5y−z = 0.Find a…

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Q: 2 Find the dimension of the subspace H of R spanned by – 12 8 ..... dim H =

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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: 4. Consider the subspace V of R' given by 1 3 V = span (a) Find an orthonormal basis for the…

A: Solution:-