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I am currently taking a proofs based linear algebra course for math undergraduates. It's been almost two years since I took a more computational linear algebra course (solving matrix equations, finding determinants, those types of things), so forgive me for how basic this question sounds.

I was wondering what properties (I was thinking of things like injective, surjective, invertible) of a linear map can be determined from its matrix? I feel like I learned this in my computational linear algebra course, but I don't remember, and I can't seem to find anything here on StackExchange or online.

Any help would be much appreciated! Thank you in advance!

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closed as too broad by GNUSupporter 8964民主女神 地下教會, Lord Shark the Unknown, Eevee Trainer, user91500, José Carlos Santos Dec 27 '18 at 11:12

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  • $\begingroup$ I don't know if this is useful, but in my class we're using Sheldon Axler's Linear Algebra Done Right $\endgroup$ – Missyinvisible Dec 10 '18 at 3:03
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A matrix $A$ represents a linear map between two vector spaces $V$ and $W$ with respect to a specific basis in each vector space. A change of basis in either or both of $V$ and $W$ transforms $A$ into an equivalent matrix

$B=Q^{-1}AP$

A property of the linear map itself (as opposed to a property of a specific matrix representation of that map) must be true of $B$ as well as of $A$. So properties of a linear map are linked to properties of a matrix that are invariant under equivalence. The main matrix property that is invariant under equivalence is its rank. From the rank of a matrix you can determine whether the corresponding linear map is injective, surjective or invertible (i.e. bijective).

If $V=W$ then $A$ is a square matrix and the linear map is an endomorphism. Equivalence is replaced by the stronger relationship of similarity:

$B = P^{-1}AP$

The characteristic polynomial of a square matrix is invariant under similarity. So this shows that attributes that can be determined from the characteristics polynomial, such as determinant, trace and eigenvalues, are properties of the underlying endomorphism, not just of the matrix representation.

On the other hand, matrix properties that are not invariant under equivalence or similarity (e.g. the number of $0$ entries in the matrix) are not properties of the underlying linear map.

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All properties. A linear map (between finite-dimensional vector spaces) is determined by its matrix.

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  • $\begingroup$ I understand how a linear map and its matrix are connected, but I was hoping to understand what the form of the matrix says about the linear map? (Like how you can tell what a linear map's eigenvalues are by looking at its matrix with respect to a basis that makes it upper triangular.) $\endgroup$ – Missyinvisible Dec 10 '18 at 3:07
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    $\begingroup$ Yes. The eigenvalues of an upper triangular matrix are the diagonal entries. $\endgroup$ – Robert Israel Dec 10 '18 at 3:30

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