Please, help me with a step by step solutions to this problem and I will be very grateful to you. I don't have a strong background in algebra. I hope you will help. Problem 6.7. Let A be any nx k matrix(1) Prove that the k x k matrix ATA and the matrix A have the same nullspace. Usethis to prove that rank(ATA) = rank(A). Similarly, prove that the n x n matrix AAT andthe matrix AT have the samenullspace, and conclude that rank(AAT) =rank(AT)We will prove later that rank(AT) = rank(A)(2) Let a1,.. ., a be k linearly independent vectors in R" (1 k

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Answered: Problem 6.7. Let A be any nx k matrix…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please, help me with a step by step solutions to this problem and I will be very grateful to you. I don't have a strong background in algebra. I hope you will help.

Problem 6.7. Let A be any nx k matrix
(1) Prove that the k x k matrix ATA and the matrix A have the same nullspace. Use
this to prove that rank(ATA) = rank(A). Similarly, prove that the n x n matrix AAT and
the matrix AT have the same
nullspace, and conclude that rank(AAT) =rank(AT)
We will prove later that rank(AT) = rank(A)
(2) Let a1,.. ., a be k linearly independent vectors in R" (1 k <n), and let A be the
n x k matrix whose ith column is aj. Prove that A A has rank k, and that it is invertible
Let P A(ATA)-1AT
an n x n matrix). Prove that
What is the matrix P when k
1?
(3) Prove that the image of P is the subspace V spanned by a1,.. . ,ak, or
the set of all vectors in R" of the form Ar, with r E R*. Prove that the nullspace U of P is
the set of vectors uE R" such that A u 0. Can you give a geometric interpretation of U?
equivalently
projection of R" onto the subspace V spanned by ai,... , ak, and
Conclude that P is a
that
R" U V

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