Mark the following either true or false. Justification is required.(a) If the homogeneous equation Ax = 0 has a nontrivial solution, then thecolumns of A are linearly independent.(b) If a set contains fewer vectors than there are entries in the vectors, thenthe set is linearly independent.(c) If a set contains more vectors than there are entries in the vectors, thenthe set is linearly dependent.

(a) Suppose that the homogenous equation has a nontrivial solution. Let the columns of are…

Answered: Mark the following either true or…

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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Question

Mark the following either true or false. Justification is required.
(a) If the homogeneous equation Ax = 0 has a nontrivial solution, then the
columns of A are linearly independent.
(b) If a set contains fewer vectors than there are entries in the vectors, then
the set is linearly independent.
(c) If a set contains more vectors than there are entries in the vectors, then
the set is linearly dependent.

Expert Solution

Step 1: ''Introduction to the solution''

(a) Suppose that the homogenous equation $A x equals 0$ has a nontrivial solution.

Let the columns of $A$ are linearly independent.

Then, $A$ has full column rank and hence the matrix $A$ is invertible that is $A to the power of negative 1 end exponent$ exists .

Then, premultiplying by $A to the power of negative 1 end exponent$ in both sides of $A x equals 0 space comma space w e space g e t comma space$ $A to the power of negative 1 end exponent open parentheses A x close parentheses equals A to the power of negative 1 end exponent.0 equals 0$

$rightwards double arrow x equals 0$ [since, $A A to the power of negative 1 end exponent equals I space right square bracket$

$rightwards double arrow A x equals 0$ has a trivial solution., a contradiction.

Thus, the columns of $A$ are linearly dependent.

Hence, the given statement(a) is false.