is the following span a basis for R^3?

1 1 1
2 , 2 , 2
3 4 10

Question

is the following span a basis for R^3?

1   1   1
2 , 2 , 2
3   4  10

kinggeorge · Answer

Column vectors?

amistre64 · Answer

span{v1,v2,v3}  yes

anonymous · Answer

i would guess not since second row is twice the first row, right?

amistre64 · Answer

i say its not, but the teacher wrote it up as an example on the board .... and claimed it was :)

anonymous · Answer

then again my linear algebra is fairly weak, so i should probably be quiet

amistre64 · Answer

|dw:1332257931259:dw|

amistre64 · Answer

im sure its a plane in R^3

anonymous · Answer

ok it is weak but not that weak. no way!

kinggeorge · Answer

Since you're using column vectors, the first vectors are obviously linearly independent since the first two elements of both are equal, but the third is different.

kinggeorge · Answer

So we just need to show that the last vector is linearly independent. This is relatively straightforward, and it just so happens to be true.

amistre64 · Answer

http://www.wolframalpha.com/input/?i=rref%7B%7B1%2C1%2C1%7D%2C%7B2%2C2%2C2%7D%2C%7B3%2C4%2C10%7D%7D

anonymous · Answer

i have to disagree, but i will watch and see

anonymous · Answer

in n by n matrix row space = column space

anonymous · Answer

I think the last column vector is linearly dependent, unless I'm mistaken.

amistre64 · Answer

i think so too

amistre64 · Answer

considering its row equivalent to a matrix the has a determinant of 0

anonymous · Answer

of course

anonymous · Answer

theorem 7 here gives it http://tutorial.math.lamar.edu/Classes/LinAlg/FundamentalSubspaces.aspx

anonymous · Answer

wolfram verifies it http://www.wolframalpha.com/input/?i=span+ {1%2C1%2C1}%2C{+2+%2C+2+%2C+2}%2C{+3+%2C4+%2C1}

anonymous · Answer

your prof was sleeping

amistre64 · Answer

he was just trying to think on his feet, thats all :)

anonymous · Answer

maybe but it is a pretty bizarre mistake to make, not that random

kinggeorge · Answer

Why did I say independent? It is dependent. So you were correct. This is not a span for $\mathbb{R}^3$

anonymous · Answer

It's a basis for $\mathbb{R}^2$, though.

amistre64 · Answer

R^2 yes; but just not the conventional xy plane

anonymous · Answer

Redefining $x,y$ whenever we see fit. :)

amistre64 · Answer

the teacher knows his stuff, but gets twisted about in the numbers alot :)

anonymous · Answer

Are you taking linear algebra?

amistre64 · Answer

yep

anonymous · Answer

If I may ask this without being rude, why are you still on this subject? Most college courses are up to projection around this time.

amistre64 · Answer

its just the roll of the cirriculum

amistre64 · Answer

he was trying to go over change of basis and colA rowA nulA and those things today