4. Which of the following statements is/are true? (A) For a nonzero matrix A E Rx, let P be a matrix that orthogonally projects vectors in R¹ onto the column space of A. Then In- P projects onto the right null space of A. (B) If A € R³x2 has two orthonormal columns, then ATA = 1₂. 00 (C) The orthogonal projection of vectors in R³ onto the column space of A == 02 can 30 000 be obtained by multiplying the vector with the projection matrix P == 001 0 1 0
4. Which of the following statements is/are true? (A) For a nonzero matrix A E Rx, let P be a matrix that orthogonally projects vectors in R¹ onto the column space of A. Then In- P projects onto the right null space of A. (B) If A € R³x2 has two orthonormal columns, then ATA = 1₂. 00 (C) The orthogonal projection of vectors in R³ onto the column space of A == 02 can 30 000 be obtained by multiplying the vector with the projection matrix P == 001 0 1 0
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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![4. Which of the following statements is/are true?
(A) For a nonzero matrix A E Rnxn, let P be a matrix that orthogonally projects vectors
in R" onto the column space of A. Then In-P projects onto the right null space of
A.
(B) If A € R³x2 has two orthonormal columns, then ATA = I₂.
0
can
(C) The orthogonal projection of vectors in R³ onto the column space of A
30
000
be obtained by multiplying the vector with the projection matrix P = 0 0 1
0 1 0
(D) Three orthogonal vectors in R4 can be constructed by the Gram-Schmidt procedure
from the vectors [1,-1,0,0], [0, 1,-1,0] and [1,0,0,-1].
(E) None of the above is true.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9f22f0f-42b5-4c22-9258-89e6890e73ac%2Fb69bd20a-9914-4442-bdb2-d8dc76c78e41%2F8s30e6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Which of the following statements is/are true?
(A) For a nonzero matrix A E Rnxn, let P be a matrix that orthogonally projects vectors
in R" onto the column space of A. Then In-P projects onto the right null space of
A.
(B) If A € R³x2 has two orthonormal columns, then ATA = I₂.
0
can
(C) The orthogonal projection of vectors in R³ onto the column space of A
30
000
be obtained by multiplying the vector with the projection matrix P = 0 0 1
0 1 0
(D) Three orthogonal vectors in R4 can be constructed by the Gram-Schmidt procedure
from the vectors [1,-1,0,0], [0, 1,-1,0] and [1,0,0,-1].
(E) None of the above is true.
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