# Thread: How to prove vectors span a space?

1. ## How to prove vectors span a space?

Hi,

How would i prove that three given vectors span the space R^4?

The vectors are as follows:

a1= [ 0, 5,0,2 ]
a2= [6,1,3,-1]
a3= [1,4,4,3]

Thanks,
Carl

2. Originally Posted by Carl505
Hi,

How would i prove that three given vectors span the space R^4?

The vectors are as follows:

a1= [ 0, 5,0,2 ]
a2= [6,1,3,-1]
a3= [1,4,4,3]

Thanks,
Carl
They don't. $\mathbb{R} ^4$ requires at least 4 vectors to span it.

-Dan

3. There is another problem in my textbook that gives three different vectors and states that the subspace of R^4 is spanned by the given vectors, and then it asks to show that theses vectors form the a basis for V (which is the subspace)

So does that mean that they are not a basis for the subspace because they do not span the the subspace? But then again, the problem states that the space is spanned by the vectors.

4. There is a difference in spanning a space such as $R^4$ and spanning a subspace of that space. If you can show that the three given vectors are linearly independent then those three vectors span a 3-dimensional subspace of $R^4$.

5. Ok that makes sense...Thanks!

one more question though,

Can you show the linear indepence by finding the row reduced echelon form of the matrix with the three vectors. And then if they have leading ones in the columns, that would conclude that they are linear independent.

Or is there another way of showing their linear independence?

6. Originally Posted by Carl505
Can you show the linear indepence by finding the row reduced echelon form of the matrix with the three vectors.
Yes, count the nonzero rows.