Eigenvalues, Eigenspaces, and Basis

[Saladsamurai]

Homework Statement

Find the eigen values, eigenspaces of the following matrix and also determine a basis for each eigen space for A = [1, 2; 3, 4]

Homework Equations

$\det(\mathbf{A} - \lambda\mathbf{I}) = 0$

The Attempt at a Solution

OK, so I found the eigenvalues and eigenspaces just fine. For eigen values I found $\lambda_{1,2} = -0.372, 5.372}$ which matches the answer in text. I also found that e₁ = [-1.457, 1]^T and e₂ = [0.4575, 1]^T which is also correct.

It is this "basis" thing that I am not understanding. Please keep in mind this is an engineering advanced math methods course, so though I do know about matrix operations, I am by no means a linear algebra wizard over here.

Upon looking up the definition of a basis, it seems that it is just a set of linearly independent (LI) vectors that can be used to write all of the other vectors in that particular set in terms of (i.e. as linear combinations of). OK great. So a set {u₁, u₂} in a vector space S is a basis for S iff u₁ and u₂ are LI. And apparently the set needs to "span S" too (still working on what that means).

So I guess what my big question is, is what is my S for which I am trying to find basises (spelling?) for? How do I start this?

 

[micromass]

So, if I understand correctly, there is an eigenvector $$e_1$$ associated with $$\lambda_1$$.
You've only found 1 vector associated with $$\lambda_1$$. There are more eigenvectors (which together make up the eigenspace), namely all multiples of the vector. Thus the eigenspace is

$$E_1=\{\alpha e_1~\vert~\alpha\in \mathbb{R}\}$$

A basis for this space is just a linearly independent set which generates $$E_1$$. Right now, you've already found a vector which generates $$E_1$$, that is: $$e_1$$ generates the space. Thus this vector is a basis: the basis of $$E_1$$ is thus $$\{e_1\}$$.