Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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(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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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.
(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)
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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)
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