Suppose A is an n × n matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. a. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivots. Yes b. If -A is not invertible, then A is also not invertible. Yes c. If the linear transformation T(x)= Ax is one-to-one, then the columns of A form a linearly dependent set. No d. If the equation Ax = 0 has the trivial solution, then the columns of A span R". Yes e. If the linear transformation T(x)= Ax is onto, then it is also one-to-one. Yes Note: saying Ax = 0 has the trivial solution does not mean that Ax = 0 necessarily has only the trivial solution.
Suppose A is an n × n matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true and sometimes false. a. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivots. Yes b. If -A is not invertible, then A is also not invertible. Yes c. If the linear transformation T(x)= Ax is one-to-one, then the columns of A form a linearly dependent set. No d. If the equation Ax = 0 has the trivial solution, then the columns of A span R". Yes e. If the linear transformation T(x)= Ax is onto, then it is also one-to-one. Yes Note: saying Ax = 0 has the trivial solution does not mean that Ax = 0 necessarily has only the trivial solution.
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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Transcribed Image Text:Suppose A is an n x n matrix. Select "yes" if the statement is always true, "no" if the statement is always false, and "maybe" if the statement is sometimes true
and sometimes false.
a. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivots.
Yes
b. If -A is not invertible, then A is also not invertible.
Yes
c. If the linear transformation T(x)= Ax is one-to-one, then the columns of A form a linearly dependent set.
No
d. If the equation Ax = 0 has the trivial solution, then the columns of A span R".
Yes
e. If the linear transformation T(x)= Ax is onto, then it is also one-to-one.
Yes
Note: saying Ax = 0 has the trivial solution does not mean that Ax = 0 necessarily has only the trivial solution.
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