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Difference between determining a transformation is linear when given vector components not points?
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I understand how to do $T(x,y,z) = (x+2,y-3,z)$
but get confused when given a vector ex. $T(v) = (vcdot w,vcdot 2w,vcdot3w)$
both are in $mathbbR^3$
Specifically, how can I prove the above transformation is linear?
linear-algebra
asked Jul 27 at 0:50
shroomyshroomy
91
 |Â
show 4 more comments
up vote
0
down vote
favorite
I understand how to do $T(x,y,z) = (x+2,y-3,z)$
but get confused when given a vector ex. $T(v) = (vcdot w,vcdot 2w,vcdot3w)$
both are in $mathbbR^3$
Specifically, how can I prove the above transformation is linear?
linear-algebra
asked Jul 27 at 0:50
shroomyshroomy
91
I'm sorry you're confused. What is your question?
– saulspatz
Jul 27 at 0:54
how do I test if this transformation is linear? I would normally use the two requirements of a linear equation and manipulate the equations to prove that they hold. I can't figure out how when given vectors and a transformation that included two vectors multiplied
– shroomyshroomy
Jul 27 at 0:56
1
The comment belongs in the body of your question.
– saulspatz
Jul 27 at 0:57
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that $w$ is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
– saulspatz
Jul 27 at 0:59
That answer belongs in the answer section.
– shroomyshroomy
Jul 27 at 1:05
 |Â
show 4 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I understand how to do $T(x,y,z) = (x+2,y-3,z)$
but get confused when given a vector ex. $T(v) = (vcdot w,vcdot 2w,vcdot3w)$
both are in $mathbbR^3$
Specifically, how can I prove the above transformation is linear?
linear-algebra
asked Jul 27 at 0:50
shroomyshroomy
91
I understand how to do $T(x,y,z) = (x+2,y-3,z)$
but get confused when given a vector ex. $T(v) = (vcdot w,vcdot 2w,vcdot3w)$
both are in $mathbbR^3$
Specifically, how can I prove the above transformation is linear?
linear-algebra
asked Jul 27 at 0:50
shroomyshroomy
91
edited Jul 27 at 1:06
asked Jul 27 at 0:50
shroomyshroomy
91
asked Jul 27 at 0:50
shroomyshroomy
91
asked Jul 27 at 0:50
shroomyshroomy
91
91
I'm sorry you're confused. What is your question?
– saulspatz
Jul 27 at 0:54
how do I test if this transformation is linear? I would normally use the two requirements of a linear equation and manipulate the equations to prove that they hold. I can't figure out how when given vectors and a transformation that included two vectors multiplied
– shroomyshroomy
Jul 27 at 0:56
1
The comment belongs in the body of your question.
– saulspatz
Jul 27 at 0:57
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that $w$ is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
– saulspatz
Jul 27 at 0:59
That answer belongs in the answer section.
– shroomyshroomy
Jul 27 at 1:05
 |Â
show 4 more comments
I'm sorry you're confused. What is your question?
– saulspatz
Jul 27 at 0:54
how do I test if this transformation is linear? I would normally use the two requirements of a linear equation and manipulate the equations to prove that they hold. I can't figure out how when given vectors and a transformation that included two vectors multiplied
– shroomyshroomy
Jul 27 at 0:56
1
The comment belongs in the body of your question.
– saulspatz
Jul 27 at 0:57
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that $w$ is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
– saulspatz
Jul 27 at 0:59
That answer belongs in the answer section.
– shroomyshroomy
Jul 27 at 1:05
I'm sorry you're confused. What is your question?
– saulspatz
Jul 27 at 0:54
I'm sorry you're confused. What is your question?
– saulspatz
Jul 27 at 0:54
how do I test if this transformation is linear? I would normally use the two requirements of a linear equation and manipulate the equations to prove that they hold. I can't figure out how when given vectors and a transformation that included two vectors multiplied
– shroomyshroomy
Jul 27 at 0:56
how do I test if this transformation is linear? I would normally use the two requirements of a linear equation and manipulate the equations to prove that they hold. I can't figure out how when given vectors and a transformation that included two vectors multiplied
– shroomyshroomy
Jul 27 at 0:56
1
1
The comment belongs in the body of your question.
– saulspatz
Jul 27 at 0:57
The comment belongs in the body of your question.
– saulspatz
Jul 27 at 0:57
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that $w$ is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
– saulspatz
Jul 27 at 0:59
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that $w$ is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
– saulspatz
Jul 27 at 0:59
That answer belongs in the answer section.
– shroomyshroomy
Jul 27 at 1:05
That answer belongs in the answer section.
– shroomyshroomy
Jul 27 at 1:05
 |Â
show 4 more comments
1 Answer
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up vote
1
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You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that w is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
answered Jul 27 at 1:20
saulspatz
10.4k21323
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that w is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
answered Jul 27 at 1:20
saulspatz
10.4k21323
add a comment |Â
up vote
1
down vote
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that w is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
answered Jul 27 at 1:20
saulspatz
10.4k21323
add a comment |Â
up vote
1
down vote
up vote
1
down vote
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that w is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
answered Jul 27 at 1:20
saulspatz
10.4k21323
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that w is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
answered Jul 27 at 1:20
saulspatz
10.4k21323
answered Jul 27 at 1:20
saulspatz
10.4k21323
answered Jul 27 at 1:20
saulspatz
10.4k21323
answered Jul 27 at 1:20
saulspatz
10.4k21323
10.4k21323
add a comment |Â
add a comment |Â
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I'm sorry you're confused. What is your question?
– saulspatz
Jul 27 at 0:54
how do I test if this transformation is linear? I would normally use the two requirements of a linear equation and manipulate the equations to prove that they hold. I can't figure out how when given vectors and a transformation that included two vectors multiplied
– shroomyshroomy
Jul 27 at 0:56
1
The comment belongs in the body of your question.
– saulspatz
Jul 27 at 0:57
You do exactly the same thing. Here, you need to use the properties of the dot product. Perhaps you haven't realized that $w$ is a constant vector. Give it a try, and if you run into trouble, edit the question to show precisely where you are having difficulty.
– saulspatz
Jul 27 at 0:59
That answer belongs in the answer section.
– shroomyshroomy
Jul 27 at 1:05