Difference between kernel of a matrix and kernel of a transformation?
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Does the word "kernel" carry different meanings in the contexts of matrices and transformations? Or are the definitions of the two in those two contexts intertwined in some way?
I know the kernel of a matrix to be it's null space. I haven't been taught about the kernel of a transformation yet, but I looked it up and it's a bunch of complicated stuff that doesn't make much sense (it's apparently every vector v that results in a transformation T outputting the zero vector?)
Are they just two different ideas?
linear-algebra matrices linear-transformations definition
edited Dec 15 '18 at 15:27
José Carlos Santos
160k22127232
asked Dec 15 '18 at 15:04
James RonaldJames Ronald
1257
$endgroup$
add a comment |
$begingroup$
Does the word "kernel" carry different meanings in the contexts of matrices and transformations? Or are the definitions of the two in those two contexts intertwined in some way?
I know the kernel of a matrix to be it's null space. I haven't been taught about the kernel of a transformation yet, but I looked it up and it's a bunch of complicated stuff that doesn't make much sense (it's apparently every vector v that results in a transformation T outputting the zero vector?)
Are they just two different ideas?
linear-algebra matrices linear-transformations definition
edited Dec 15 '18 at 15:27
José Carlos Santos
160k22127232
asked Dec 15 '18 at 15:04
James RonaldJames Ronald
1257
$endgroup$
$begingroup$
See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
$endgroup$
– Mauro ALLEGRANZA
Dec 15 '18 at 15:10
add a comment |
$begingroup$
Does the word "kernel" carry different meanings in the contexts of matrices and transformations? Or are the definitions of the two in those two contexts intertwined in some way?
I know the kernel of a matrix to be it's null space. I haven't been taught about the kernel of a transformation yet, but I looked it up and it's a bunch of complicated stuff that doesn't make much sense (it's apparently every vector v that results in a transformation T outputting the zero vector?)
Are they just two different ideas?
linear-algebra matrices linear-transformations definition
edited Dec 15 '18 at 15:27
José Carlos Santos
160k22127232
asked Dec 15 '18 at 15:04
James RonaldJames Ronald
1257
$endgroup$
Does the word "kernel" carry different meanings in the contexts of matrices and transformations? Or are the definitions of the two in those two contexts intertwined in some way?
I know the kernel of a matrix to be it's null space. I haven't been taught about the kernel of a transformation yet, but I looked it up and it's a bunch of complicated stuff that doesn't make much sense (it's apparently every vector v that results in a transformation T outputting the zero vector?)
Are they just two different ideas?
linear-algebra matrices linear-transformations definition
linear-algebra matrices linear-transformations definition
edited Dec 15 '18 at 15:27
José Carlos Santos
160k22127232
asked Dec 15 '18 at 15:04
James RonaldJames Ronald
1257
edited Dec 15 '18 at 15:27
José Carlos Santos
160k22127232
asked Dec 15 '18 at 15:04
James RonaldJames Ronald
1257
edited Dec 15 '18 at 15:27
José Carlos Santos
160k22127232
edited Dec 15 '18 at 15:27
José Carlos Santos
160k22127232
edited Dec 15 '18 at 15:27
José Carlos Santos
160k22127232
160k22127232
asked Dec 15 '18 at 15:04
James RonaldJames Ronald
1257
asked Dec 15 '18 at 15:04
James RonaldJames Ronald
1257
asked Dec 15 '18 at 15:04
James RonaldJames Ronald
1257
1257
$begingroup$
See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
$endgroup$
– Mauro ALLEGRANZA
Dec 15 '18 at 15:10
add a comment |
$begingroup$
See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
$endgroup$
– Mauro ALLEGRANZA
Dec 15 '18 at 15:10
$begingroup$
See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
$endgroup$
– Mauro ALLEGRANZA
Dec 15 '18 at 15:10
$begingroup$
See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
$endgroup$
– Mauro ALLEGRANZA
Dec 15 '18 at 15:10
add a comment |
2 Answers
2
active
oldest
votes
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The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.
edited Dec 15 '18 at 15:18
Ivo Terek
46.2k953142
answered Dec 15 '18 at 15:10
José Carlos SantosJosé Carlos Santos
160k22127232
$endgroup$
add a comment |
$begingroup$
Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.
answered Dec 15 '18 at 15:32
user247327user247327
11.1k1515
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add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
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active
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$begingroup$
The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.
edited Dec 15 '18 at 15:18
Ivo Terek
46.2k953142
answered Dec 15 '18 at 15:10
José Carlos SantosJosé Carlos Santos
160k22127232
$endgroup$
add a comment |
$begingroup$
The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.
edited Dec 15 '18 at 15:18
Ivo Terek
46.2k953142
answered Dec 15 '18 at 15:10
José Carlos SantosJosé Carlos Santos
160k22127232
$endgroup$
add a comment |
$begingroup$
The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.
edited Dec 15 '18 at 15:18
Ivo Terek
46.2k953142
answered Dec 15 '18 at 15:10
José Carlos SantosJosé Carlos Santos
160k22127232
$endgroup$
The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.
edited Dec 15 '18 at 15:18
Ivo Terek
46.2k953142
answered Dec 15 '18 at 15:10
José Carlos SantosJosé Carlos Santos
160k22127232
edited Dec 15 '18 at 15:18
Ivo Terek
46.2k953142
edited Dec 15 '18 at 15:18
Ivo Terek
46.2k953142
edited Dec 15 '18 at 15:18
Ivo Terek
46.2k953142
46.2k953142
answered Dec 15 '18 at 15:10
José Carlos SantosJosé Carlos Santos
160k22127232
answered Dec 15 '18 at 15:10
José Carlos SantosJosé Carlos Santos
160k22127232
answered Dec 15 '18 at 15:10
José Carlos SantosJosé Carlos Santos
160k22127232
160k22127232
add a comment |
add a comment |
$begingroup$
Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.
answered Dec 15 '18 at 15:32
user247327user247327
11.1k1515
$endgroup$
add a comment |
$begingroup$
Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.
answered Dec 15 '18 at 15:32
user247327user247327
11.1k1515
$endgroup$
add a comment |
$begingroup$
Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.
answered Dec 15 '18 at 15:32
user247327user247327
11.1k1515
$endgroup$
Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.
answered Dec 15 '18 at 15:32
user247327user247327
11.1k1515
answered Dec 15 '18 at 15:32
user247327user247327
11.1k1515
answered Dec 15 '18 at 15:32
user247327user247327
11.1k1515
answered Dec 15 '18 at 15:32
user247327user247327
11.1k1515
11.1k1515
add a comment |
add a comment |
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$begingroup$
See Kernel of a Linear transformation : The kernel of a linear transformation L is the set of all vectors v such that $ L(v) = 0$.
$endgroup$
– Mauro ALLEGRANZA
Dec 15 '18 at 15:10