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$$\{(1,0,0)^T, (0,1,1)^T, (1,0,1)^T, (1,2,3)^T\}$$

What I did first was write the linear combination of these vectors which I do not know how to format on here but is simply:

$$c_1(1,0,0) + c_2(0,1,1) + c_3(1,0,1) + c_4(1,2,3) = (x, y, z)$$

where the vectors are column vectors.

Then I get the equations $c_1 + c_3 + c_4 = x, c_2 + 2c_4 = y, c_2 + c_3 + 3c_4 = z$.

And this is where I get stuck. How do I prove that the vectors span $\Bbb R^3$ from here?

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A simple way to prove a span would be to show how each of the three standard basis vectors can me made as a linear combination of your set. –  Henning Makholm Oct 2 '12 at 14:16

2 Answers 2

In fact you have that the first three vectors listed span all of $\mathbb{R}^3$. So in your style you would have

$$c_1(1,0,0) + c_2(0,1,1) + c_3(1,0,1)= (x, y, z)$$ implying that $$\begin{align} c_1 + c_3 &= x \\ c_2 &= y\\ c_2 + c_3 &= z. \end{align} $$

This system of equations you should be able to solve.

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Hint: If you can generate $[1,0,0]^T$, $[0,1,0]^T$ and $[0,0,1]^T$ with these vectors, then you can span the whole space.

Using this it's easy to see that just the first three vectors span the space.

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+1 as it also answers the question how to determine subspace spanned by a set of vectors in general. –  smihael Jul 9 '13 at 14:09

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