Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

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?

share|cite|improve this question
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

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.

share|cite|improve this answer
+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

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.