(a) If A is an m × n matrix and R is the reduced echelon form of A, chen col(A) = col(R). (Here, col(A) means the column space of A.)
(a) If A is an m × n matrix and R is the reduced echelon form of A, chen col(A) = col(R). (Here, col(A) means the column space of A.)
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 answer with True or False, provide evidence as well.
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![(a) If A is an m × n matrix and R is the reduced echelon form of A,
then col(A) = col (R). (Here, col(A) means the column space of A.)
(b) If A is an m x n matrix and R is the reduced echelon form of A,
then dim(col(A))
dim(col(R)).
(c) If matrices A and B are row equivalent, then row(A) = row(B).
(Here, row(A) means the row space of A.)
(d) If A is a 6 × 8 matrix with exactly 2 non-pivot columns, then the
pivot columns of A form a basis of R6.
(e) If A is a 6 x 8 matrix such that dim(row(A)) = 5, then the nullity
of Atr equals 2. (Here, row(A) means the row space of A.)
(f) If B = (vi, v2, V3, V4) is a basis for some vector space V, then the
coordinate vector of v with respect to the basis B is [v4]B
1
(3)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F091efdc9-e15b-4ccc-8207-a2051acfb2c9%2Fc3278da8-6def-47a0-970d-d588b811395a%2F9g49pqr_processed.png&w=3840&q=75)
Transcribed Image Text:(a) If A is an m × n matrix and R is the reduced echelon form of A,
then col(A) = col (R). (Here, col(A) means the column space of A.)
(b) If A is an m x n matrix and R is the reduced echelon form of A,
then dim(col(A))
dim(col(R)).
(c) If matrices A and B are row equivalent, then row(A) = row(B).
(Here, row(A) means the row space of A.)
(d) If A is a 6 × 8 matrix with exactly 2 non-pivot columns, then the
pivot columns of A form a basis of R6.
(e) If A is a 6 x 8 matrix such that dim(row(A)) = 5, then the nullity
of Atr equals 2. (Here, row(A) means the row space of A.)
(f) If B = (vi, v2, V3, V4) is a basis for some vector space V, then the
coordinate vector of v with respect to the basis B is [v4]B
1
(3)
![(g) If B = (vi, v2, 03, 04) and B' = (w1,w2, w3, WA) are two bases for
some vector space V, then the transition matrices Pg-→B' and Pg-→B
satisfy the following property:
[Ps¬B][Pg ¬B] = I4,
where [A][B] means the product of matrices A and B and I4 is the 4x 4
identity matrix.
(h) If S is the standard basis of R³ and B = (vi, v2, v3) is another basis
of R°, then the transition matrix Pg-s has the following form: its first
column is the vector vi, its second column is the vector v2, and its third
column is the vector v3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F091efdc9-e15b-4ccc-8207-a2051acfb2c9%2Fc3278da8-6def-47a0-970d-d588b811395a%2Fta4gcos_processed.png&w=3840&q=75)
Transcribed Image Text:(g) If B = (vi, v2, 03, 04) and B' = (w1,w2, w3, WA) are two bases for
some vector space V, then the transition matrices Pg-→B' and Pg-→B
satisfy the following property:
[Ps¬B][Pg ¬B] = I4,
where [A][B] means the product of matrices A and B and I4 is the 4x 4
identity matrix.
(h) If S is the standard basis of R³ and B = (vi, v2, v3) is another basis
of R°, then the transition matrix Pg-s has the following form: its first
column is the vector vi, its second column is the vector v2, and its third
column is the vector v3.
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