Question

Are the columns of A linearly independent? Why?

Answer 1

\[\left[\begin{matrix}1 & -1 &2 &1 \\ 2 & -3 &2&0 \\-1&1&2&3\\-3&2&0&3\end{matrix}\right]\]

Answer 2

Do I have to find the determinant?

Answer 3

That would be the most straightforward way, but if you look closely you can tell just by looking.

Answer 4

How so?

Answer 5

You can see that the fourth column is actually the third minus the first. But generally speaking the determinant is the best way to tell.

Answer 6

oh okay thank you

Answer 7

just wondering if there be any difference between the phrases "linearly independent columns" versus "linearly independent rows"
in both cases determinant comes 0.... just thinking if there is any difference...

Answer 8

rows, refers to equations directly.. so its but intuitive
but columns refers to linear combinations of these equations or wat... i feel lost a bit :|

Answer 9

Linearly dependent rows means that one equation is a combination of the other equations, while linearly dependent columns means that one variable is a combination of the other variables.

Answer 10

ohk, that makes sense a bit,
can we have linearly dependent rows , w/o having linearly dependent columns ?

Answer 11

we can get from one to other by row transformations right

Answer 12

I'll need to revise i guess, thanks @Jemurray3 :)

Answer 13

Oops, I made an error in the above explanation so I deleted it. To rephrase, no you cannot have one without the other, because linearly dependent columns implies linearly dependent rows.

Answer 14

Ok uh.. I did out the determinant thing so I could make sure but I got 1? It's supposed to be 0 isn't it

Answer 15

okie il give it further thought @Jemurray3 , thanks for the time :D

Answer 16

im getting det = 0..

Answer 17

Yes, it should be zero, recheck your work.

Answer 18

oh yup! Lil mistake. Thanks!

Answer 19

you may get check these things... just to be confident from wolfram
http://www.wolframalpha.com/input/?i=determinant+of+%7B%7B1+%2C+-1+%2C2+%2C1%7D%2C+%7B2+%2C+-3+%2C2%2C0%7D%2C%7B-1%2C1%2C2%2C3%7D%2C+%7B-3%2C2%2C0%2C3%7D%7D