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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Asked Sep 28, 2019
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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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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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Expert Answer

Step 1

Since we only answer up to first question, we’ll answer the first 1. Please resubmit the question and specify the other squestions (up to 1) you’d like answered.

1.A is the matrix of order n*k and tr(A). A is the matrix of order k*k.

    Null space of matrix A is defined as {x : x  belongs to R(n-tuple) and Ax=0}

    Such that N(A)= {x : x  belongs to R(n-tuple) and Ax=0}

    Consider any x which belongs to null apace of A. Then

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Ax 0 right multiply by A in the above equation AT Ax A (0) =0 Which implies that x e NAA Hence, N(A) N(ATA ...1)

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Step 2

Now, consider any x which belongs to null space of tr(A). A, then

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A Ax 0 left multiply by x in the above equation Ax(Ax(Ax) 0 (Since length of the vector Ax is 0) Hence, N(AA) N(A) (2)

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Step 3

By equations (1) and (2) it is clear that null space A and n...

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N(AA) N(A)

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