Suppose that A is an m × n matrix and that B is an n x p matrix. (a) im(A) related?) Show that rank(AB) < rank(A). (Hint: How are the subspaces im(AB) and (b) to part (a) to show that the 3 x 3 matrix AB cannot be invertible. Suppose that A is a 3 × 2 matrix and B is a 2 × 3 matrix. Use your answer (c) Show that dim ker(B) < dim ker(AB). In addition to what was shown in part (a) of this problem, it is also true (d) that rank(AB) < rank(B). Use the result of part (c), together with the rank-nullity theorem, to show this fact.
Suppose that A is an m × n matrix and that B is an n x p matrix. (a) im(A) related?) Show that rank(AB) < rank(A). (Hint: How are the subspaces im(AB) and (b) to part (a) to show that the 3 x 3 matrix AB cannot be invertible. Suppose that A is a 3 × 2 matrix and B is a 2 × 3 matrix. Use your answer (c) Show that dim ker(B) < dim ker(AB). In addition to what was shown in part (a) of this problem, it is also true (d) that rank(AB) < rank(B). Use the result of part (c), together with the rank-nullity theorem, to show this fact.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:4. Suppose that A is an m × n matrix and that B is an n × p matrix.
(a)
im(A) related?)
Show that rank(AB) < rank(A). (Hint: How are the subspaces im(AB) and
(b)
to part (a) to show that the 3 × 3 matrix AB cannot be invertible.
Suppose that A is a 3 × 2 matrix and B is a 2 × 3 matrix. Use your answer
(c)
Show that dim ker(B) < dim ker(AB).
In addition to what was shown in part (a) of this problem, it is also true
(d)
that rank(AB) < rank(B). Use the result of part (c), together with the rank-nullity
theorem, to show this fact.1
Expert Solution
Step 1 matrix = linear map
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
