Question

Determine whether the vectors in the set S span the vector space V.
V = R^2 S = ([0,0], [1,1])

Answer 1

What do you think?

Answer 2

I dont know how to do it

Answer 3

What does it mean to span a vector space?

Answer 4

idk

Answer 5

One thing tha can help is that the identity matrix will span its appropriate dimensional vector space. So a 2x2 identity matrix will span R^2, a 3x3 identity matrix will span r^3, etc.

Answer 6

so it does span?

Answer 7

Can your set of vectors S be written or transformed into a 2x2 identity matrix?

Answer 8

[0 1]
[0 1]

Answer 9

Right. That is not a 2x2 identity matrix nor will it ever be. A guaranteed set of vectors that will span a vector space are those column vectors that from the identity matrix.

In order to span a space, you need to be able to have a set of vectors such that a linear combination of those vectors can be used to represent every vector in that space. So in order to span R^2, I need to be able to generate EVERY column vector of the form \[\left(\begin{matrix}R \\ R\end{matrix}\right)\] where R is any real number.

The problem with your set of vectors is you have this:
\(\left(\begin{matrix}0 \\ 0\end{matrix}\right) \) and \(\left(\begin{matrix}1 \\ 1\end{matrix}\right) \). There is no linbear combination of these vectors that could give me the vector \(\left(\begin{matrix}1 \\ 2\end{matrix}\right) \) for example. So there's no way your set could span all of R^2, if that makes sense.

Answer 10

what do you mean by linear combination giving you 1 and 2?

Answer 11

Like, for any numbers a and b, can I ever have

\(a\)\(\left(\begin{matrix}0 \\ 0\end{matrix}\right) \) + \(b\)\(\left(\begin{matrix}1 \\ 1\end{matrix}\right) \) = \(\left(\begin{matrix}1 \\ 2\end{matrix}\right) \)

??