Linear Operator , visual meaning.
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Could someone give me the visual intuition, if there is any, of a Linear Operator $L$ ?
Given the definition that $L$ is linear if:
${L(f+g)=Lf+Lg}$ (with f and g being functions)
and
${L(tf)=tL(f)}$ (with t being scalar)
linear-algebra abstract-algebra measure-theory
edited Nov 17 at 20:47
mathreadler
14.6k72160
asked Nov 17 at 14:37
Ricouello
1355
add a comment |
up vote
0
down vote
favorite
Could someone give me the visual intuition, if there is any, of a Linear Operator $L$ ?
Given the definition that $L$ is linear if:
${L(f+g)=Lf+Lg}$ (with f and g being functions)
and
${L(tf)=tL(f)}$ (with t being scalar)
linear-algebra abstract-algebra measure-theory
edited Nov 17 at 20:47
mathreadler
14.6k72160
asked Nov 17 at 14:37
Ricouello
1355
What do you mean by "visual meaning" ?
– Rebellos
Nov 17 at 14:39
Like a visual intuition, if I have an image on a graph and apply a linear operator to it, how does it changes and in which way?
– Ricouello
Nov 17 at 14:48
Go in the plane and see what happens there. It can be a rotation, dilation, shearing, reflection or something else
– Fakemistake
Nov 17 at 15:12
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Could someone give me the visual intuition, if there is any, of a Linear Operator $L$ ?
Given the definition that $L$ is linear if:
${L(f+g)=Lf+Lg}$ (with f and g being functions)
and
${L(tf)=tL(f)}$ (with t being scalar)
linear-algebra abstract-algebra measure-theory
edited Nov 17 at 20:47
mathreadler
14.6k72160
asked Nov 17 at 14:37
Ricouello
1355
Could someone give me the visual intuition, if there is any, of a Linear Operator $L$ ?
Given the definition that $L$ is linear if:
${L(f+g)=Lf+Lg}$ (with f and g being functions)
and
${L(tf)=tL(f)}$ (with t being scalar)
linear-algebra abstract-algebra measure-theory
linear-algebra abstract-algebra measure-theory
edited Nov 17 at 20:47
mathreadler
14.6k72160
asked Nov 17 at 14:37
Ricouello
1355
edited Nov 17 at 20:47
mathreadler
14.6k72160
asked Nov 17 at 14:37
Ricouello
1355
edited Nov 17 at 20:47
mathreadler
14.6k72160
edited Nov 17 at 20:47
mathreadler
14.6k72160
edited Nov 17 at 20:47
mathreadler
14.6k72160
14.6k72160
asked Nov 17 at 14:37
Ricouello
1355
asked Nov 17 at 14:37
Ricouello
1355
asked Nov 17 at 14:37
Ricouello
1355
1355
What do you mean by "visual meaning" ?
– Rebellos
Nov 17 at 14:39
Like a visual intuition, if I have an image on a graph and apply a linear operator to it, how does it changes and in which way?
– Ricouello
Nov 17 at 14:48
Go in the plane and see what happens there. It can be a rotation, dilation, shearing, reflection or something else
– Fakemistake
Nov 17 at 15:12
add a comment |
What do you mean by "visual meaning" ?
– Rebellos
Nov 17 at 14:39
Like a visual intuition, if I have an image on a graph and apply a linear operator to it, how does it changes and in which way?
– Ricouello
Nov 17 at 14:48
Go in the plane and see what happens there. It can be a rotation, dilation, shearing, reflection or something else
– Fakemistake
Nov 17 at 15:12
What do you mean by "visual meaning" ?
– Rebellos
Nov 17 at 14:39
What do you mean by "visual meaning" ?
– Rebellos
Nov 17 at 14:39
Like a visual intuition, if I have an image on a graph and apply a linear operator to it, how does it changes and in which way?
– Ricouello
Nov 17 at 14:48
Like a visual intuition, if I have an image on a graph and apply a linear operator to it, how does it changes and in which way?
– Ricouello
Nov 17 at 14:48
Go in the plane and see what happens there. It can be a rotation, dilation, shearing, reflection or something else
– Fakemistake
Nov 17 at 15:12
Go in the plane and see what happens there. It can be a rotation, dilation, shearing, reflection or something else
– Fakemistake
Nov 17 at 15:12
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
Operator is a name for a special kind of function. Recall that in a geometric context functions are called maps: sloppily speaking a map sends points to points. Not all maps are linear, take a translation, e.g.
Now an operator is a function whose domain are functions itself: usually it sends functions to functions. Not all operators are linear, but the example of the differential operator R.Burton gave is linear. The term "Linear Operator" is often used as a name for linear mappings between infinite-dimensional vector spaces.
To make another example: Let $V:=mathbb R^{mathbb N}$ the vector space of all real-valued sequences. Then define the linear operator $Scolon Vto V$, defined by $Sbigl((a_1, a_2,a_3,dotsc)bigr)=(0,a_1,a_2,a_3,dotsc)$.
answered Nov 17 at 18:18
Michael Hoppe
10.6k31733
add a comment |
up vote
1
down vote
The wikipedia article on linear maps has a pretty good animation. Outside of the elementary example, the next easiest case would probably be the differential operator $D(f(x))=frac{d}{dx}f(x)$. I would recommend using Desmos or GeoGebra to play with different functions and see how they behave under the action of different operators. It's a bit more specific, but you could also check out 3blue1brown's video on group theory.
answered Nov 17 at 17:51
R. Burton
1267
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Operator is a name for a special kind of function. Recall that in a geometric context functions are called maps: sloppily speaking a map sends points to points. Not all maps are linear, take a translation, e.g.
Now an operator is a function whose domain are functions itself: usually it sends functions to functions. Not all operators are linear, but the example of the differential operator R.Burton gave is linear. The term "Linear Operator" is often used as a name for linear mappings between infinite-dimensional vector spaces.
To make another example: Let $V:=mathbb R^{mathbb N}$ the vector space of all real-valued sequences. Then define the linear operator $Scolon Vto V$, defined by $Sbigl((a_1, a_2,a_3,dotsc)bigr)=(0,a_1,a_2,a_3,dotsc)$.
answered Nov 17 at 18:18
Michael Hoppe
10.6k31733
add a comment |
up vote
1
down vote
accepted
Operator is a name for a special kind of function. Recall that in a geometric context functions are called maps: sloppily speaking a map sends points to points. Not all maps are linear, take a translation, e.g.
Now an operator is a function whose domain are functions itself: usually it sends functions to functions. Not all operators are linear, but the example of the differential operator R.Burton gave is linear. The term "Linear Operator" is often used as a name for linear mappings between infinite-dimensional vector spaces.
To make another example: Let $V:=mathbb R^{mathbb N}$ the vector space of all real-valued sequences. Then define the linear operator $Scolon Vto V$, defined by $Sbigl((a_1, a_2,a_3,dotsc)bigr)=(0,a_1,a_2,a_3,dotsc)$.
answered Nov 17 at 18:18
Michael Hoppe
10.6k31733
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Operator is a name for a special kind of function. Recall that in a geometric context functions are called maps: sloppily speaking a map sends points to points. Not all maps are linear, take a translation, e.g.
Now an operator is a function whose domain are functions itself: usually it sends functions to functions. Not all operators are linear, but the example of the differential operator R.Burton gave is linear. The term "Linear Operator" is often used as a name for linear mappings between infinite-dimensional vector spaces.
To make another example: Let $V:=mathbb R^{mathbb N}$ the vector space of all real-valued sequences. Then define the linear operator $Scolon Vto V$, defined by $Sbigl((a_1, a_2,a_3,dotsc)bigr)=(0,a_1,a_2,a_3,dotsc)$.
answered Nov 17 at 18:18
Michael Hoppe
10.6k31733
Operator is a name for a special kind of function. Recall that in a geometric context functions are called maps: sloppily speaking a map sends points to points. Not all maps are linear, take a translation, e.g.
Now an operator is a function whose domain are functions itself: usually it sends functions to functions. Not all operators are linear, but the example of the differential operator R.Burton gave is linear. The term "Linear Operator" is often used as a name for linear mappings between infinite-dimensional vector spaces.
To make another example: Let $V:=mathbb R^{mathbb N}$ the vector space of all real-valued sequences. Then define the linear operator $Scolon Vto V$, defined by $Sbigl((a_1, a_2,a_3,dotsc)bigr)=(0,a_1,a_2,a_3,dotsc)$.
answered Nov 17 at 18:18
Michael Hoppe
10.6k31733
edited Nov 17 at 18:27
answered Nov 17 at 18:18
Michael Hoppe
10.6k31733
answered Nov 17 at 18:18
Michael Hoppe
10.6k31733
answered Nov 17 at 18:18
Michael Hoppe
10.6k31733
10.6k31733
add a comment |
add a comment |
up vote
1
down vote
The wikipedia article on linear maps has a pretty good animation. Outside of the elementary example, the next easiest case would probably be the differential operator $D(f(x))=frac{d}{dx}f(x)$. I would recommend using Desmos or GeoGebra to play with different functions and see how they behave under the action of different operators. It's a bit more specific, but you could also check out 3blue1brown's video on group theory.
answered Nov 17 at 17:51
R. Burton
1267
add a comment |
up vote
1
down vote
The wikipedia article on linear maps has a pretty good animation. Outside of the elementary example, the next easiest case would probably be the differential operator $D(f(x))=frac{d}{dx}f(x)$. I would recommend using Desmos or GeoGebra to play with different functions and see how they behave under the action of different operators. It's a bit more specific, but you could also check out 3blue1brown's video on group theory.
answered Nov 17 at 17:51
R. Burton
1267
add a comment |
up vote
1
down vote
up vote
1
down vote
The wikipedia article on linear maps has a pretty good animation. Outside of the elementary example, the next easiest case would probably be the differential operator $D(f(x))=frac{d}{dx}f(x)$. I would recommend using Desmos or GeoGebra to play with different functions and see how they behave under the action of different operators. It's a bit more specific, but you could also check out 3blue1brown's video on group theory.
answered Nov 17 at 17:51
R. Burton
1267
The wikipedia article on linear maps has a pretty good animation. Outside of the elementary example, the next easiest case would probably be the differential operator $D(f(x))=frac{d}{dx}f(x)$. I would recommend using Desmos or GeoGebra to play with different functions and see how they behave under the action of different operators. It's a bit more specific, but you could also check out 3blue1brown's video on group theory.
answered Nov 17 at 17:51
R. Burton
1267
answered Nov 17 at 17:51
R. Burton
1267
answered Nov 17 at 17:51
R. Burton
1267
answered Nov 17 at 17:51
R. Burton
1267
1267
add a comment |
add a comment |
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What do you mean by "visual meaning" ?
– Rebellos
Nov 17 at 14:39
Like a visual intuition, if I have an image on a graph and apply a linear operator to it, how does it changes and in which way?
– Ricouello
Nov 17 at 14:48
Go in the plane and see what happens there. It can be a rotation, dilation, shearing, reflection or something else
– Fakemistake
Nov 17 at 15:12