# Thread: Spanning sets for R^3

1. ## Spanning sets for R^3

Need to show that {(1,0,0)^T, (0,1,1)^T, (1,0,1)^T, (1,2,3)^T} is a spanning set for R^3.

So I use alpha, beta, gamma, and another variable to show that this entire statement can be put into a linear combination. Would I just set the last variable to be equal to 0. It seems otherwise it would be a spanning set of R^4 instead of R^3.

2. Originally Posted by pakman
Need to show that {(1,0,0)^T, (0,1,1)^T, (1,0,1)^T, (1,2,3)^T} is a spanning set for R^3.
Consider $\textbf{u} =\begin{pmatrix} x\\ y\\ z \end{pmatrix}\in \mathbb{R}^3$.We will show this general vector can be obtained by a linear combination of the first 3 elements. To do that we have to obtain the scalars of the combination.

Originally Posted by pakman
So I use alpha, beta, gamma, and another variable to show that this entire statement can be put into a linear combination.
$\textbf{u} = a\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} + b\begin{pmatrix} 0\\ 1\\ 1 \end{pmatrix}+c\begin{pmatrix} 1\\ 0\\ 1 \end{pmatrix}$

$x = a+b$
$y = b$
$z = b+c$
This means $(a,b,c) = (x-y,y,z-y)$. So we have the scalars for the linear combination.

Originally Posted by pakman
Would I just set the last variable to be equal to 0.
No you cant directly do that. You should prove the first three elements are linearly independent. And then claim this set is adequate using the fact that the dimension of the space is 3.

Originally Posted by pakman
It seems otherwise it would be a spanning set of R^4 instead of R^3
If a spanning set has 4 vectors, the dimension of the basis it spans need not be 4. I can put 100 more vectors in that spanning set. But that set shall still span $\mathbb{R}^3$. Dont forget that whether it span $\mathbb{R}^3$ or $\mathbb{R}^4$ is decided by the maximum number of linearly independent vectors in the set.

P.S: Also remember that $\mathbb{R}^3$ has 3-tuples and $\mathbb{R}^4$ has 4-tuples

3. Thanks, that helped a lot. I'm confused about one thing you said though, what is a 3-tuple?

4. Originally Posted by pakman
Thanks, that helped a lot. I'm confused about one thing you said though, what is a 3-tuple?
Numbers like $(1,2,3)$ or $(e,233344,\pi)$. Basically vectors with 3 co-ordinates