Welcome to PF.
I recommend Michael Artin's book: Algebra, as a good guide to linear algebra and other topics in algebra. With your indulgence, I take this opportunity to suggest a long list of pure math topics, in algebra, calculus, complex analysis, topology, and number theory, going up through grad school level:

Some basic pure math results, with references:

root- factor theorem for polynomials
rational root theorem
euclidean algorithm
fundamental theorem of arithmetic (unique prime factorization
Quadratic reciprocity
[Euclid: Elements, Book 7, Prop.1,2; Shifrin, Abstract algebra, a geometric approach; Gauss Disquisitiones, section II, par. 16, section IV, par. 131]

fundamental theorem of linear algebra (rank/nullity thm)
Row reduction/echelon form (Gaussian elimination)
Cayley-Hamilton theorem
rational canonical/jordan form
decomposition of finite abelian groups as products
[M. Artin: Algebra]

fundamental theorem of calculus
Fubini theorem
Green/Stokes theorem
Inverse/implicit function theorem
Existence of solutions of first order differential equations
[Spivak : Calculus, Calculus on manifolds, Differential geometry, vol.1; Lang, Analysis I (new title: Undergraduate analysis) ]

Cauchy integral theorem/formula
Fundamental theorem of algebra
Riemann-Roch theorem
[Cartan, Elementary theory of analytic functions; Fulton: Algebraic topology, a first course]

Fundamental theorem of Galois theory
Criterion for solvability of polynomials via radicals
[E. Artin: Galois theory]

Fundamental theorem of covering spaces/fundamental group
Poincare duality for homology of manifolds
Jordan curve theorem
Poincare’-Hopf (index of vector fields)
Hurewicz isomorphism for homotopy/homology
[Massey: Algebraic topology, an introduction; Fulton: algebraic topology, a first course; Guillemin-Pollack: Differential topology; Milnor: Topology from the differentiable viewpoint; Bott-Tu: Differential forms in algebraic topology]

Dirichlet theorem on primes in arithmetic progressions
[Serre: A course in arithmetic]

In addition to these expositions by experts, which I most highly recommend, some of my class notes are available free here:
https://www.math.uga.edu/directory/people/roy-smith

E.g. here is one my several sets of notes on linear algebra:
https://www.math.uga.edu/sites/default/files/laprimexp.pdf

Specifically for linear algebra, here is an especially good, and cheap, book by Shilov, in my opinion.
https://www.amazon.com/dp/048663518X?tag=pfamazon01-20