Section 3.4: Number 8 8. Let W denote the subspace of R5 consisting of all vectors having coor-dinates that sum to zero. The vectorsu1 (2, 3, 4, -5, 2),=--5,2),u3 (3,-2, 7, -9, 1),ԱՅ=u5 (-1, 1, 2, 1, -3),=u2 (-6, 9, 12, 15, -6),=-u4 (2, -8,2,-2, 6),U6=(0, -3, -18, 9, 12),=u7 (1,0, 2, 3,-2),andUg =(2,-1, 1, -9,7)generate W. Find a subset of {u₁, u2,..., us} that is a basis for W.

Answered: Image /qna-images/answer/4b81d501-11c0-406a-ac1d-f358106d231c.jpg

Answered: Let W denote the subspace of R5…

Elementary Linear Algebra (MindTap Course List)

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Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 54CR: Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector...

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Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
determine whether the first vector is in the subspace of R3 generated by the other vectors.1) (1, 4, -3); (1, 0, 1), (1, 1, 0), (3, 1, 2)2) (1, 1, 2); (0, 1, 0), (3, 5, 6), (1, 2, 1)
A set of 4 vectors in R5 is linearly independent cannot span R5 spans a 4-dimensional subspace of R5
Determine which of the following subspaces of R3are identical: U1= span[ (1,1,-1), (2,3,-1), (3,1,-5)], U2= span[ (1,-1,-3), (3,-2,-8), (2,1,-3)], U3= span[ (1,1,1), (1,-1,3), (3,-1,7)]
Find a basis for the subspace of R4 spanned by S. S = {(2, 5, −3, −3), (−2, −3, 2, −2), (1, 3, −2, 3), (−1, −5, 3, 2)}
There are four vectors (1,a,b,c), (1,1,1,1), (1,2,3,1), (3,4,5,3). If the vector space (subspace) consisting of the linear combination of these vectors is T, find the condition that the number of vectors constituting the basis of T will be 3.
Show that the set of all solutions to the equation Ax = 0, where A = [1 2 3 -2 -1 0] [1 2 2 -1 -2 1] [2 4 5 -3 -2 0] forms a vector subspace in R^6. Find three special solutions corresponding to different free columns.
Find the dimension of the subspace W of R4 spanned by S = {v1, v2, v3} = {(−1, 2, 5, 0), (3, 0, 1, −2), (−5, 4, 9, 2)}.
Let W denote the subspace of R5 consisting of all vectors having coordinates that sum to zero. The vectors u1= (2,−3, 4,−5, 2), u2 = (−6, 9,−12, 15,−6),u3 = (3,−2, 7,−9, 1), u4 = (2,−8, 2,−2, 6),u5= (−1, 1, 2, 1,−3), u6 = (0,−3,-18, 9, 12),u7 = (1, 0,−2, 3,−2),and u8 = (2,−1, 1,−9, 7)generate W. Find a subset of {u1, u2, . . . ,u8} that is a basis for W.
Why are [0,1] and (0,1] which are subspaces of the real number system with the usual topology not homeomorphic?

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