What properties of a linear map can be determined from its matrix? [closed]
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!
linear-algebra matrices linear-transformations big-list
edited Dec 10 '18 at 13:36
user 170039
10.4k42465
asked Dec 10 '18 at 2:59
Missyinvisible
353
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
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
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!
linear-algebra matrices linear-transformations big-list
edited Dec 10 '18 at 13:36
user 170039
10.4k42465
asked Dec 10 '18 at 2:59
Missyinvisible
353
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
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
I don't know if this is useful, but in my class we're using Sheldon Axler's Linear Algebra Done Right
– Missyinvisible
Dec 10 '18 at 3:03
add a comment |
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!
linear-algebra matrices linear-transformations big-list
edited Dec 10 '18 at 13:36
user 170039
10.4k42465
asked Dec 10 '18 at 2:59
Missyinvisible
353
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!
linear-algebra matrices linear-transformations big-list
linear-algebra matrices linear-transformations big-list
edited Dec 10 '18 at 13:36
user 170039
10.4k42465
asked Dec 10 '18 at 2:59
Missyinvisible
353
edited Dec 10 '18 at 13:36
user 170039
10.4k42465
asked Dec 10 '18 at 2:59
Missyinvisible
353
edited Dec 10 '18 at 13:36
user 170039
10.4k42465
edited Dec 10 '18 at 13:36
user 170039
10.4k42465
edited Dec 10 '18 at 13:36
user 170039
10.4k42465
10.4k42465
asked Dec 10 '18 at 2:59
Missyinvisible
353
asked Dec 10 '18 at 2:59
Missyinvisible
353
asked Dec 10 '18 at 2:59
Missyinvisible
353
353
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
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
I don't know if this is useful, but in my class we're using Sheldon Axler's Linear Algebra Done Right
– Missyinvisible
Dec 10 '18 at 3:03
add a comment |
I don't know if this is useful, but in my class we're using Sheldon Axler's Linear Algebra Done Right
– Missyinvisible
Dec 10 '18 at 3:03
I don't know if this is useful, but in my class we're using Sheldon Axler's Linear Algebra Done Right
– Missyinvisible
Dec 10 '18 at 3:03
I don't know if this is useful, but in my class we're using Sheldon Axler's Linear Algebra Done Right
– Missyinvisible
Dec 10 '18 at 3:03
add a comment |
2 Answers
2
active
oldest
votes
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.
answered Dec 10 '18 at 14:34
gandalf61
7,693623
add a comment |
All properties. A linear map (between finite-dimensional vector spaces) is determined by its matrix.
answered Dec 10 '18 at 3:03
Robert Israel
318k23208457
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.)
– Missyinvisible
Dec 10 '18 at 3:07
1
Yes. The eigenvalues of an upper triangular matrix are the diagonal entries.
– Robert Israel
Dec 10 '18 at 3:30
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered Dec 10 '18 at 14:34
gandalf61
7,693623
add a comment |
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.
answered Dec 10 '18 at 14:34
gandalf61
7,693623
add a comment |
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.
answered Dec 10 '18 at 14:34
gandalf61
7,693623
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.
answered Dec 10 '18 at 14:34
gandalf61
7,693623
answered Dec 10 '18 at 14:34
gandalf61
7,693623
answered Dec 10 '18 at 14:34
gandalf61
7,693623
answered Dec 10 '18 at 14:34
gandalf61
7,693623
7,693623
add a comment |
add a comment |
All properties. A linear map (between finite-dimensional vector spaces) is determined by its matrix.
answered Dec 10 '18 at 3:03
Robert Israel
318k23208457
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.)
– Missyinvisible
Dec 10 '18 at 3:07
1
Yes. The eigenvalues of an upper triangular matrix are the diagonal entries.
– Robert Israel
Dec 10 '18 at 3:30
add a comment |
All properties. A linear map (between finite-dimensional vector spaces) is determined by its matrix.
answered Dec 10 '18 at 3:03
Robert Israel
318k23208457
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.)
– Missyinvisible
Dec 10 '18 at 3:07
1
Yes. The eigenvalues of an upper triangular matrix are the diagonal entries.
– Robert Israel
Dec 10 '18 at 3:30
add a comment |
All properties. A linear map (between finite-dimensional vector spaces) is determined by its matrix.
answered Dec 10 '18 at 3:03
Robert Israel
318k23208457
All properties. A linear map (between finite-dimensional vector spaces) is determined by its matrix.
answered Dec 10 '18 at 3:03
Robert Israel
318k23208457
answered Dec 10 '18 at 3:03
Robert Israel
318k23208457
answered Dec 10 '18 at 3:03
Robert Israel
318k23208457
answered Dec 10 '18 at 3:03
Robert Israel
318k23208457
318k23208457
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.)
– Missyinvisible
Dec 10 '18 at 3:07
1
Yes. The eigenvalues of an upper triangular matrix are the diagonal entries.
– Robert Israel
Dec 10 '18 at 3:30
add a comment |
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.)
– Missyinvisible
Dec 10 '18 at 3:07
1
Yes. The eigenvalues of an upper triangular matrix are the diagonal entries.
– Robert Israel
Dec 10 '18 at 3:30
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.)
– Missyinvisible
Dec 10 '18 at 3:07
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.)
– Missyinvisible
Dec 10 '18 at 3:07
1
1
Yes. The eigenvalues of an upper triangular matrix are the diagonal entries.
– Robert Israel
Dec 10 '18 at 3:30
Yes. The eigenvalues of an upper triangular matrix are the diagonal entries.
– Robert Israel
Dec 10 '18 at 3:30
add a comment |
I don't know if this is useful, but in my class we're using Sheldon Axler's Linear Algebra Done Right
– Missyinvisible
Dec 10 '18 at 3:03