Linear Algebra Terminology Trouble I have been reading up on how to bring a matrix to diagonal form, and I learned that a matrix is diagonalizable if and only if the eigenvectors "span the space." What does it mean for eigenvectors to "span the space"? I am inferring  that it means to "fill in" the the columns of a matrix $\textbf{S}^{-1}$ that diagonalizes $\textbf{T}$  using the eigenvectors of $\textbf{T}$. For example, if a matrix $\textbf{T}$  has two colums, it will need two linearly independent eigenvectors to "span the space." Thank you. 

We say that a set of vectors $\{v_1,\ldots, v_n\}\in V$ spans the finite dimensional vector space $V$ if every vector $w\in V$ can be written in the form 
$$\sum_{i=1}^n a_iv_i=w $$
where here $a_1,\ldots, a_n$ are elements of the coefficient field $\mathbb{F}$. $\mathbb{F}$ is most likely $\mathbb{R}$ or $\mathbb{C}$ for your purposes. Basically, the notion of a spanning set boils down to being able to express any vector in the space in terms of combinations of the vectors in the set. For instance, the vector $1$ spans $\mathbb{R}$, because any $k\in \mathbb{R}$ satisfies $k=k*1$. Similarly, $(1,1), (1,0),(0,1)$ spans $\mathbb{R}^2$ because any $(a,b)\in \mathbb{R}^2$ can be written as a combination of $(1,1),(1,0),(0,1)$, as you can check.
