How do you find the basis for the spanning set which contains matrices?

[lawnmowjob]

Please look at the link: http://gyazo.com/5f57caddf7dc76aa61d387f2915d29fe.png

 

[Simon_Tyler]

Not all of the matrices in your link are independenthttp://www.wolframalpha.com/input/?...1}}+++d*{{2,+1},+{1,+2}}+==+{{0,+0},+{0,+0}}" Find the largest independent set and that is your basis.

 

[HallsofIvy]

Given a set of vectors that spans a vector space, the largest subset that contains only independent vectors is a basis. The condition for dependence is that
$$a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n= 0$$
with at least on of the scalars non-zero. If it is, say [\itex]a_i\ne 0[/itex], we can solve for the vector $v_i$ as a function of the others and so can drop it from the set.

Here, that equation is
$$a_1\begin{bmatrix}0& 3\\ 1 & 1\end{bmatrix}+ a_2\begin{bmatrix}4 & 5 \\ 3 & 1 \end{bmatrix}+ a_3\begin{bmatrix}-4 & 1 \\ -1 & 1\end{bmatrix}+ a_4\begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix}$$
$$= \begin{bmatrix}4a_2- 4a_3+ 2a_4 & 3a_1+ 5a_2+ a_3+ a_4 \\ a_1+ 3a_2- a_3+ a_4 & a_1+ a_2+ a_3+ 2a_4\end{bmatrix}$$
which is the same as the four equations
[tex]4a_2- 4a_3+ 2a_4= 0[/itex]
$$3a_1+ 5a_2+ a_3+ a_4= 0$$
$$a_1+ 3a_2- a_3+ a_4= 0$$
$$a_1+ a_2+ a_3+ 2a_4= 0$$

An obvious solution is $a_1= a_2= a_3= a_4= 0$. If that is the only solution, the four matrices are independent and so are a basis for there span. If there is a solution in which one of the coefficients is non-zero, we can solve for that matrix in terms of the other three and so drop it from the set.

 

[Simon_Tyler]

@HallsofIvy: I suspect that lawnmowjob asked this as a homework problem, then either figured it out or got help from somewhere else. Given the shortness of his original post, I doubt that he'll be back. It happens too often in these forums, and makes me reluctant to write out more explicit and helpful discussions like yours.

Anyway, I hid the answer in the middle full stop of my answer.

 

[HallsofIvy]

Cute! I'll have to remember that!

 

[Deveno]

sad, but true. many people just want "the answer", and have little interest in gaining the knowledge.