I have an assignment where I should proof that three matrices span $\mathbb{R}^{3\times 1}$. Here is the matrices:
$$ b_1=\pmatrix{1\\ 1\\ 1\\} b_2=\pmatrix{0\\1\\1\\} b_3=\pmatrix{0\\-1\\1}$$
For this I have to determine this two things:
$$c_1\pmatrix{1\\ 1\\ 1\\} +c_2\pmatrix{0\\1\\1\\} +c_3\pmatrix{0\\-1\\1} = \pmatrix{0\\0\\0\\}$$
And:
$$c_1\pmatrix{1\\ 1\\ 1\\} +c_2\pmatrix{0\\1\\1\\} +c_3\pmatrix{0\\-1\\1} = \pmatrix{a\\b\\c\\}$$
So if I do some matrix algebra I get this:
$$\pmatrix{c_1\\c_1+c_2-c_3\\c_1+c_2+c_3\\}=\pmatrix{0\\0\\0\\}$$
And:
$$\pmatrix{c_1\\c_1+c_2-c_3\\c_1+c_2+c_3\\}=\pmatrix{a\\b\\c\\}$$
And then what? Am I done now? In case of that, how could this not be true?
Best Answer
An easy way is to determine the rank of the matrix $$\pmatrix{1 \ \ \ \ 0 \ \ \ \ 0 \\ 1 \ \ \ 1 \ \ -1\\ 1\ \ \ \ 1 \ \ \ \ 1\\} $$ If the rank of this matrix is 3, then this set of vectos span $\mathbb{R}^3.$