4. Let U be the subspace of Mnxn consisting of all A Є Mnxn such that=aij 0 for all i, j E• •{1,2,dimension of U (you do not need to prove that U is a subspace).., n} with i > j, where aij denotes the entry of A located at row i, column j. Find the

Finding the DimensionTo find the dimension of U, we need to determine the number of linearly…

Answered: 4. Let U be the subspace of Mnxn…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 43EQ

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Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.
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.
Let A be an nn matrix in which the entries of each row sum to zero. Find |A|.
Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the closest to u that is, show that d(u,projvu) is a minimum.
8. Let W denote the subspace of R5 consisting of all the vectors having coordinates that sum to zero. The vectors u₁=(2, -3,4,5, 2), = uz (3,-2,7,-9, 1), u5 (-1, 1, 2, 1, -3), u7 (1,0, 2, 3,-2), U5 = = u2=(-6, 9, -12, 15, -6), u4=(2,-8, 2, -2, 6), U6 = u6 (0, -3, -18, 9, 12), us=(2, -1, 1,-9, 7) generate W. Find a subset of the set {ui, u2,..., us} that is a basis for W.
Let W be the subspace of IR' spanned by the vectors 1 4 -2 and -1 -7 Find the matrix A of the orthogonal projection onto W. A

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4. Let U be the subspace of Mnxn consisting of all A Є Mnxn such that = aij 0 for all i, j E • • {1,2, dimension of U (you do not need to prove that U is a subspace). ., n} with i > j, where aij denotes the entry of A located at row i, column j. Find the