Linear Operator clarification
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Could someone explain why f is not a linear operator while g is?
What are the criteria to distinguish if a function is a linear operator?
Is linear transformations another way to call for linear operators? Our book is not clear on this.
Thanks in advance.
linear-algebra functional-analysis linear-transformations
asked Nov 17 at 15:40
Ricouello
1355
add a comment |
up vote
0
down vote
favorite
Could someone explain why f is not a linear operator while g is?
What are the criteria to distinguish if a function is a linear operator?
Is linear transformations another way to call for linear operators? Our book is not clear on this.
Thanks in advance.
linear-algebra functional-analysis linear-transformations
asked Nov 17 at 15:40
Ricouello
1355
1
Does $f$ preserve linear combinations?
– Sean Roberson
Nov 17 at 15:52
Linear means : $L(vec v+vec w)=L(vec v)+L(vec w)$ and $L(cvec v)=cL(vec v)$.
– lulu
Nov 17 at 15:57
What does it mean to preserve linear combination? Like the combination of vectors multiplied by scalars used to define vectors in the same space?
– Ricouello
Nov 17 at 16:00
And that's why ab is not part of the same space in f?
– Ricouello
Nov 17 at 16:01
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Could someone explain why f is not a linear operator while g is?
What are the criteria to distinguish if a function is a linear operator?
Is linear transformations another way to call for linear operators? Our book is not clear on this.
Thanks in advance.
linear-algebra functional-analysis linear-transformations
asked Nov 17 at 15:40
Ricouello
1355
Could someone explain why f is not a linear operator while g is?
What are the criteria to distinguish if a function is a linear operator?
Is linear transformations another way to call for linear operators? Our book is not clear on this.
Thanks in advance.
linear-algebra functional-analysis linear-transformations
linear-algebra functional-analysis linear-transformations
asked Nov 17 at 15:40
Ricouello
1355
asked Nov 17 at 15:40
Ricouello
1355
asked Nov 17 at 15:40
Ricouello
1355
asked Nov 17 at 15:40
Ricouello
1355
asked Nov 17 at 15:40
Ricouello
1355
1355
1
Does $f$ preserve linear combinations?
– Sean Roberson
Nov 17 at 15:52
Linear means : $L(vec v+vec w)=L(vec v)+L(vec w)$ and $L(cvec v)=cL(vec v)$.
– lulu
Nov 17 at 15:57
What does it mean to preserve linear combination? Like the combination of vectors multiplied by scalars used to define vectors in the same space?
– Ricouello
Nov 17 at 16:00
And that's why ab is not part of the same space in f?
– Ricouello
Nov 17 at 16:01
add a comment |
1
Does $f$ preserve linear combinations?
– Sean Roberson
Nov 17 at 15:52
Linear means : $L(vec v+vec w)=L(vec v)+L(vec w)$ and $L(cvec v)=cL(vec v)$.
– lulu
Nov 17 at 15:57
What does it mean to preserve linear combination? Like the combination of vectors multiplied by scalars used to define vectors in the same space?
– Ricouello
Nov 17 at 16:00
And that's why ab is not part of the same space in f?
– Ricouello
Nov 17 at 16:01
1
1
Does $f$ preserve linear combinations?
– Sean Roberson
Nov 17 at 15:52
Does $f$ preserve linear combinations?
– Sean Roberson
Nov 17 at 15:52
Linear means : $L(vec v+vec w)=L(vec v)+L(vec w)$ and $L(cvec v)=cL(vec v)$.
– lulu
Nov 17 at 15:57
Linear means : $L(vec v+vec w)=L(vec v)+L(vec w)$ and $L(cvec v)=cL(vec v)$.
– lulu
Nov 17 at 15:57
What does it mean to preserve linear combination? Like the combination of vectors multiplied by scalars used to define vectors in the same space?
– Ricouello
Nov 17 at 16:00
What does it mean to preserve linear combination? Like the combination of vectors multiplied by scalars used to define vectors in the same space?
– Ricouello
Nov 17 at 16:00
And that's why ab is not part of the same space in f?
– Ricouello
Nov 17 at 16:01
And that's why ab is not part of the same space in f?
– Ricouello
Nov 17 at 16:01
add a comment |
1 Answer
1
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up vote
1
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The difference between linear operator and linear transformation is vague and depends on your course. Some instructors may want to distinguish these notions, while some won't. One of the common approaches is to say that linear operator is a particular case of linear transformation, when the same vector space is both domain and codomain. You may see this question for more details.
A linear transformation from vector space $V$ into vector space $W$ (over the same field $F$) is a function $T$ from $V$ into $W$ such that $forall alpha,betain V$ and $forall cin F$:
$$T(calpha+beta) = c(Talpha)+Tbeta.$$
This is just a definition that you should be able to find in any linear algebra book.
Apply the definition to check whether $f$ is a linear transformation or not. Let $v_1=(a_1,b_1)$ and $v_2=(a_2,b_2)$, then
$$f(v_1+v_2)=(a_1+a_2)cdot(b_1+b_2)+1neq (a_1cdot b_1+1)+(a_2cdot b_2+1)=f(v_1)+f(v_2),$$
thus $f$ is not a linear transformation.
N.B. Although it isn't in your question, I want to take an opportunity to highlight another common mistake and point out that linear transformation is not the same as linear function. Indeed, a linear transformation is a function that necessarily passes through the origin, because of $T(0) = T(0 + 0) = T(0) + T(0) = 0$, while arbitrary linear function doesn't have this property.
answered Nov 17 at 16:09
Hasek
1,092517
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
The difference between linear operator and linear transformation is vague and depends on your course. Some instructors may want to distinguish these notions, while some won't. One of the common approaches is to say that linear operator is a particular case of linear transformation, when the same vector space is both domain and codomain. You may see this question for more details.
A linear transformation from vector space $V$ into vector space $W$ (over the same field $F$) is a function $T$ from $V$ into $W$ such that $forall alpha,betain V$ and $forall cin F$:
$$T(calpha+beta) = c(Talpha)+Tbeta.$$
This is just a definition that you should be able to find in any linear algebra book.
Apply the definition to check whether $f$ is a linear transformation or not. Let $v_1=(a_1,b_1)$ and $v_2=(a_2,b_2)$, then
$$f(v_1+v_2)=(a_1+a_2)cdot(b_1+b_2)+1neq (a_1cdot b_1+1)+(a_2cdot b_2+1)=f(v_1)+f(v_2),$$
thus $f$ is not a linear transformation.
N.B. Although it isn't in your question, I want to take an opportunity to highlight another common mistake and point out that linear transformation is not the same as linear function. Indeed, a linear transformation is a function that necessarily passes through the origin, because of $T(0) = T(0 + 0) = T(0) + T(0) = 0$, while arbitrary linear function doesn't have this property.
answered Nov 17 at 16:09
Hasek
1,092517
add a comment |
up vote
1
down vote
The difference between linear operator and linear transformation is vague and depends on your course. Some instructors may want to distinguish these notions, while some won't. One of the common approaches is to say that linear operator is a particular case of linear transformation, when the same vector space is both domain and codomain. You may see this question for more details.
A linear transformation from vector space $V$ into vector space $W$ (over the same field $F$) is a function $T$ from $V$ into $W$ such that $forall alpha,betain V$ and $forall cin F$:
$$T(calpha+beta) = c(Talpha)+Tbeta.$$
This is just a definition that you should be able to find in any linear algebra book.
Apply the definition to check whether $f$ is a linear transformation or not. Let $v_1=(a_1,b_1)$ and $v_2=(a_2,b_2)$, then
$$f(v_1+v_2)=(a_1+a_2)cdot(b_1+b_2)+1neq (a_1cdot b_1+1)+(a_2cdot b_2+1)=f(v_1)+f(v_2),$$
thus $f$ is not a linear transformation.
N.B. Although it isn't in your question, I want to take an opportunity to highlight another common mistake and point out that linear transformation is not the same as linear function. Indeed, a linear transformation is a function that necessarily passes through the origin, because of $T(0) = T(0 + 0) = T(0) + T(0) = 0$, while arbitrary linear function doesn't have this property.
answered Nov 17 at 16:09
Hasek
1,092517
add a comment |
up vote
1
down vote
up vote
1
down vote
The difference between linear operator and linear transformation is vague and depends on your course. Some instructors may want to distinguish these notions, while some won't. One of the common approaches is to say that linear operator is a particular case of linear transformation, when the same vector space is both domain and codomain. You may see this question for more details.
A linear transformation from vector space $V$ into vector space $W$ (over the same field $F$) is a function $T$ from $V$ into $W$ such that $forall alpha,betain V$ and $forall cin F$:
$$T(calpha+beta) = c(Talpha)+Tbeta.$$
This is just a definition that you should be able to find in any linear algebra book.
Apply the definition to check whether $f$ is a linear transformation or not. Let $v_1=(a_1,b_1)$ and $v_2=(a_2,b_2)$, then
$$f(v_1+v_2)=(a_1+a_2)cdot(b_1+b_2)+1neq (a_1cdot b_1+1)+(a_2cdot b_2+1)=f(v_1)+f(v_2),$$
thus $f$ is not a linear transformation.
N.B. Although it isn't in your question, I want to take an opportunity to highlight another common mistake and point out that linear transformation is not the same as linear function. Indeed, a linear transformation is a function that necessarily passes through the origin, because of $T(0) = T(0 + 0) = T(0) + T(0) = 0$, while arbitrary linear function doesn't have this property.
answered Nov 17 at 16:09
Hasek
1,092517
The difference between linear operator and linear transformation is vague and depends on your course. Some instructors may want to distinguish these notions, while some won't. One of the common approaches is to say that linear operator is a particular case of linear transformation, when the same vector space is both domain and codomain. You may see this question for more details.
A linear transformation from vector space $V$ into vector space $W$ (over the same field $F$) is a function $T$ from $V$ into $W$ such that $forall alpha,betain V$ and $forall cin F$:
$$T(calpha+beta) = c(Talpha)+Tbeta.$$
This is just a definition that you should be able to find in any linear algebra book.
Apply the definition to check whether $f$ is a linear transformation or not. Let $v_1=(a_1,b_1)$ and $v_2=(a_2,b_2)$, then
$$f(v_1+v_2)=(a_1+a_2)cdot(b_1+b_2)+1neq (a_1cdot b_1+1)+(a_2cdot b_2+1)=f(v_1)+f(v_2),$$
thus $f$ is not a linear transformation.
N.B. Although it isn't in your question, I want to take an opportunity to highlight another common mistake and point out that linear transformation is not the same as linear function. Indeed, a linear transformation is a function that necessarily passes through the origin, because of $T(0) = T(0 + 0) = T(0) + T(0) = 0$, while arbitrary linear function doesn't have this property.
answered Nov 17 at 16:09
Hasek
1,092517
answered Nov 17 at 16:09
Hasek
1,092517
answered Nov 17 at 16:09
Hasek
1,092517
answered Nov 17 at 16:09
Hasek
1,092517
1,092517
add a comment |
add a comment |
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Does $f$ preserve linear combinations?
– Sean Roberson
Nov 17 at 15:52
Linear means : $L(vec v+vec w)=L(vec v)+L(vec w)$ and $L(cvec v)=cL(vec v)$.
– lulu
Nov 17 at 15:57
What does it mean to preserve linear combination? Like the combination of vectors multiplied by scalars used to define vectors in the same space?
– Ricouello
Nov 17 at 16:00
And that's why ab is not part of the same space in f?
– Ricouello
Nov 17 at 16:01