The first step is to see linear algebra as encoded into vector bundles. Essentially, vector bundles are vector spaces parameterized by a topological space (or an algebraic variety). Then Serre-Swan's theorem gives a correspondence between projective modules over a commutative ring and vector bundles over a compact space (or affine variety over an algebraically closed field). This gives you classical algebraic K-theory. If you continue down this path the connection to vector spaces becomes more of a stretch; for example Waldhausen's algebraic K-theory takes as input a category with cofibrations and weak equivalences (topologically useful classes of morphisms) and gives a spectrum as output. This seems to live only in a topological world, but classical algebraic K-theory of a commutative ring can be realized as a special case.

Also, perhaps some spectra can be thought of as living in some kind of generalized linear algebraic world as well. This seems to be what Hopkins and Smith are saying in Nilpotence and Stable Homotopy Theory II

The description of spectra as cell complexes encourages the intuition that the endomorphism rings of finite spectra approximate matrix algebras over the ring $\pi_*(S^0)$.

In this paper they describe the Morava K-theory spectra $K(n)$, which have coefficients $K(n)_*\cong \mathbb{F}_p[v_n^{\pm 1}]$. These $K(n)$ act a lot like fields and this leads to some nice properties. In fact the $K(n)$ can be thought of as the prime fields.given that a ring spectrum $E$ is a field (i.e. $E\wedge X$ has the homotopy type of a wedge of suspensions of $X$) then $E$ has the homotopy type of a wedge of suspensions of $K(n)$ for some $n$. Perhaps the Morava K-theories are far afield from what you had in mind, but I think there is a common thread of linear algebraic intuition behind any form of K-theory. I think this is an interesting question (one that I have been asking as well) and I am curious what other people have to say about this.

The first step is to see linear algebra as encoded into vector bundles. Essentially, vector bundles are vector spaces parameterized by a topological space (or an algebraic variety). Then Serre-Swan's theorem gives a correspondence between projective modules over a commutative ring and vector bundles over a compact space (or affine variety over an algebraically closed field). This gives you classical algebraic K-theory. If you continue down this path the connection to vector spaces becomes more of a stretch; for example Waldhausen's algebraic K-theory takes as input a category with cofibrations and weak equivalences (topologically useful classes of morphisms) and gives a spectrum as output. This seems to live only in a topological world, but classical algebraic K-theory of a commutative ring can be realized as a special case.

Though perhaps some spectra can be thought of as living in some kind of generalized linear algebraic world as well. This seems to be what Hopkins and Smith are saying in Nilpotence and Stable Homotopy Theory II

The description of spectra as cell complexes encourages the intuition that the endomorphism rings of finite spectra approximate matrix algebras over the ring $\pi_*(S^0)$.

In this paper they describe the Morava K-theory spectra $K(n)$, which have coefficients $K(n)_*\cong \mathbb{F}_p[v_n^{\pm 1}]$. These $K(n)$ act a lot like fields and this leads to some nice properties. In fact, the $K(n)$ can be thought of as the prime fields. Given that a ring spectrum $E$ is a field (i.e. $E\wedge X$ has the homotopy type of a wedge of suspensions of $X$) then $E$ has the homotopy type of a wedge of suspensions of $K(n)$ for some $n$. Perhaps the Morava K-theories are far afield from what you had in mind, but I think there is a common thread of linear algebraic intuition behind any form of K-theory. 

I think this is an interesting question (one that I have been asking as well) and I am curious what other people have to say about this.