Question

A linear system is consistent iff there are free variables in the row-echelon form of the corresponding augmented matrix. True or false? give a counterexample if it's false
Please, help

Answer 1

hmm, a free variable represents at least a row of zeros ...

Answer 2

can form parallel lines, and augment?

Answer 3

3 5 | 7
9 17 | 7

how does that pan out?

Answer 4

http://www.wolframalpha.com/input/?i=rref%7B%7B3%2C5%2C7%7D%2C%7B9%2C15%2C7%7D%7D

Answer 5

I am sorry, the wolfram is for ... what?

Answer 6

5*3 = 15, not 17 in the previous post :)

Answer 7

it shows the rref of the augmenting of the system of parallel lines

Answer 8

we have a free variable denoted by the 2nd coumn

Answer 9

is that a counterexample?

Answer 10

does it meet the criteria?

Answer 11

yup, I am with you then

Answer 12

good :)

Answer 13

but the system \[\left[\begin{matrix}1&2\\3&1\end{matrix}\right]\]has no free variable is consists still, so the statement is not true, right?

Answer 14

@amistre64

Answer 15

A linear system is consistent
-if and only if-
there are free variables in the row-echelon
form of the corresponding augmented matrix.

we showed that a system of parallel lines (therefore not consistent) has a free variable in the rref augmented form.

Answer 16

an iff statement is true "both ways" so the hypothesis-conclusion setup can be reversed and still has to hold true

Answer 17

A iff B

if A, then B
and
if B, then A

otherwise, false

Answer 18

I got you for the way we prove iff,
if A , then B
but from if B , then A, I give a counter example above to say that " not just only B, there is a C can get A" . Is it not the way to prove?

Answer 19

unless im reading you incorrectly, you seem to be trying to insert and extra part into the setup that does not exist to begin with. There is no room for an inclusion of C into the setup.

Answer 20

i am alive, iff, i am breathing

if, i am breathing; then, i am alive.
if, i am alive; then, i am breathing.

including a third component is not needed to express the validity of the statement.

Answer 21

I am sorry, I am too worry for my test, I try to see the problem in many aspects to completely understand it.
I got you , just focus on if A then B and if B then A. That's it

Answer 22

the simplest counter-example is just as effective as the most complex ;)

Answer 23

I am distracted by "only"

Answer 24

to me, it means only B satisfy A, so counter example is not just only B. XD

Answer 25

:) yeah, its not refering to a closed system of happenstances. It just focusing on the necessary relationship between the given events.

Answer 26

most definitions are of the form iff. It is a stronger relationship than the one directional if,then construction

Answer 27

Thanks for explanation.

Answer 28

I got it.

Answer 29

good luck ;)

Answer 30

ty