Determine a linear operator $T : Bbb R^3 → Bbb R^4$ the image is generated by (2, 1, 1) and (1, -1, 2). [on...
Determine a linear operator $T:Bbb R^3 → Bbb R^4$ the image is generated by (2, 1, 1) and (1, -1, 2).
linear-algebra
asked Jan 4 at 14:11
Lidy MonteiroLidy Monteiro
72
put on hold as off-topic by StubbornAtom, zipirovich, Hans Lundmark, jgon, Cesareo Jan 5 at 0:29
This question appears to be off-topic. The users who voted to close gave this specific reason:
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Determine a linear operator $T:Bbb R^3 → Bbb R^4$ the image is generated by (2, 1, 1) and (1, -1, 2).
linear-algebra
asked Jan 4 at 14:11
Lidy MonteiroLidy Monteiro
72
put on hold as off-topic by StubbornAtom, zipirovich, Hans Lundmark, jgon, Cesareo Jan 5 at 0:29
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – StubbornAtom, zipirovich, jgon, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
3
The codomain is four-dimensional, and therefore the image must be generated by four-dimensional vectors.
– Arthur
Jan 4 at 14:13
I do not understand difference between linear operator and linear transformation. So I'm just asking for help.
– Lidy Monteiro
Jan 4 at 14:26
There is no difference. Linear map is also sometimes used. Means the same thing.
– Arthur
Jan 4 at 14:28
add a comment |
Determine a linear operator $T:Bbb R^3 → Bbb R^4$ the image is generated by (2, 1, 1) and (1, -1, 2).
linear-algebra
asked Jan 4 at 14:11
Lidy MonteiroLidy Monteiro
72
Determine a linear operator $T:Bbb R^3 → Bbb R^4$ the image is generated by (2, 1, 1) and (1, -1, 2).
linear-algebra
linear-algebra
asked Jan 4 at 14:11
Lidy MonteiroLidy Monteiro
72
asked Jan 4 at 14:11
Lidy MonteiroLidy Monteiro
72
asked Jan 4 at 14:11
Lidy MonteiroLidy Monteiro
72
asked Jan 4 at 14:11
Lidy MonteiroLidy Monteiro
72
asked Jan 4 at 14:11
Lidy MonteiroLidy Monteiro
72
72
put on hold as off-topic by StubbornAtom, zipirovich, Hans Lundmark, jgon, Cesareo Jan 5 at 0:29
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – StubbornAtom, zipirovich, jgon, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as off-topic by StubbornAtom, zipirovich, Hans Lundmark, jgon, Cesareo Jan 5 at 0:29
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – StubbornAtom, zipirovich, jgon, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
3
The codomain is four-dimensional, and therefore the image must be generated by four-dimensional vectors.
– Arthur
Jan 4 at 14:13
I do not understand difference between linear operator and linear transformation. So I'm just asking for help.
– Lidy Monteiro
Jan 4 at 14:26
There is no difference. Linear map is also sometimes used. Means the same thing.
– Arthur
Jan 4 at 14:28
add a comment |
3
The codomain is four-dimensional, and therefore the image must be generated by four-dimensional vectors.
– Arthur
Jan 4 at 14:13
I do not understand difference between linear operator and linear transformation. So I'm just asking for help.
– Lidy Monteiro
Jan 4 at 14:26
There is no difference. Linear map is also sometimes used. Means the same thing.
– Arthur
Jan 4 at 14:28
3
3
The codomain is four-dimensional, and therefore the image must be generated by four-dimensional vectors.
– Arthur
Jan 4 at 14:13
The codomain is four-dimensional, and therefore the image must be generated by four-dimensional vectors.
– Arthur
Jan 4 at 14:13
I do not understand difference between linear operator and linear transformation. So I'm just asking for help.
– Lidy Monteiro
Jan 4 at 14:26
I do not understand difference between linear operator and linear transformation. So I'm just asking for help.
– Lidy Monteiro
Jan 4 at 14:26
There is no difference. Linear map is also sometimes used. Means the same thing.
– Arthur
Jan 4 at 14:28
There is no difference. Linear map is also sometimes used. Means the same thing.
– Arthur
Jan 4 at 14:28
add a comment |
1 Answer
1
active
oldest
votes
There is a difference between 'linear operator' and 'linear transformation'. The term 'linear operator' is generally used to mean a linear transformation from a vector space to the same vector space. For example, $I:Vto V$ is a linear operator (and transformation) while $M:Bbb R^2toBbb R^3$ is only a linear transformation.
The question probably meant $T:Bbb R^3toBbb R^3$ seeing as how it mentions 'linear operator' and proceeds to give three dimensional basis vectors for the range of $T$.
Recall that the range of a linear transformation is the column space of the matrix that represents it. Thus, you want the columns of the matrix of $T,[T]$ to be linear combinations of $(2,1,1),(1,-1,2)$. For example, $$[T]=begin{bmatrix}2&1&2\1&-1&1\1&2&1end{bmatrix}implies T(x,y,z)=(2x+y+2z,x-y+z,x+2y+z)$$
There are infinitely many answers.
answered Jan 4 at 16:55
Shubham JohriShubham Johri
4,524717
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
There is a difference between 'linear operator' and 'linear transformation'. The term 'linear operator' is generally used to mean a linear transformation from a vector space to the same vector space. For example, $I:Vto V$ is a linear operator (and transformation) while $M:Bbb R^2toBbb R^3$ is only a linear transformation.
The question probably meant $T:Bbb R^3toBbb R^3$ seeing as how it mentions 'linear operator' and proceeds to give three dimensional basis vectors for the range of $T$.
Recall that the range of a linear transformation is the column space of the matrix that represents it. Thus, you want the columns of the matrix of $T,[T]$ to be linear combinations of $(2,1,1),(1,-1,2)$. For example, $$[T]=begin{bmatrix}2&1&2\1&-1&1\1&2&1end{bmatrix}implies T(x,y,z)=(2x+y+2z,x-y+z,x+2y+z)$$
There are infinitely many answers.
answered Jan 4 at 16:55
Shubham JohriShubham Johri
4,524717
add a comment |
There is a difference between 'linear operator' and 'linear transformation'. The term 'linear operator' is generally used to mean a linear transformation from a vector space to the same vector space. For example, $I:Vto V$ is a linear operator (and transformation) while $M:Bbb R^2toBbb R^3$ is only a linear transformation.
The question probably meant $T:Bbb R^3toBbb R^3$ seeing as how it mentions 'linear operator' and proceeds to give three dimensional basis vectors for the range of $T$.
Recall that the range of a linear transformation is the column space of the matrix that represents it. Thus, you want the columns of the matrix of $T,[T]$ to be linear combinations of $(2,1,1),(1,-1,2)$. For example, $$[T]=begin{bmatrix}2&1&2\1&-1&1\1&2&1end{bmatrix}implies T(x,y,z)=(2x+y+2z,x-y+z,x+2y+z)$$
There are infinitely many answers.
answered Jan 4 at 16:55
Shubham JohriShubham Johri
4,524717
add a comment |
There is a difference between 'linear operator' and 'linear transformation'. The term 'linear operator' is generally used to mean a linear transformation from a vector space to the same vector space. For example, $I:Vto V$ is a linear operator (and transformation) while $M:Bbb R^2toBbb R^3$ is only a linear transformation.
The question probably meant $T:Bbb R^3toBbb R^3$ seeing as how it mentions 'linear operator' and proceeds to give three dimensional basis vectors for the range of $T$.
Recall that the range of a linear transformation is the column space of the matrix that represents it. Thus, you want the columns of the matrix of $T,[T]$ to be linear combinations of $(2,1,1),(1,-1,2)$. For example, $$[T]=begin{bmatrix}2&1&2\1&-1&1\1&2&1end{bmatrix}implies T(x,y,z)=(2x+y+2z,x-y+z,x+2y+z)$$
There are infinitely many answers.
answered Jan 4 at 16:55
Shubham JohriShubham Johri
4,524717
There is a difference between 'linear operator' and 'linear transformation'. The term 'linear operator' is generally used to mean a linear transformation from a vector space to the same vector space. For example, $I:Vto V$ is a linear operator (and transformation) while $M:Bbb R^2toBbb R^3$ is only a linear transformation.
The question probably meant $T:Bbb R^3toBbb R^3$ seeing as how it mentions 'linear operator' and proceeds to give three dimensional basis vectors for the range of $T$.
Recall that the range of a linear transformation is the column space of the matrix that represents it. Thus, you want the columns of the matrix of $T,[T]$ to be linear combinations of $(2,1,1),(1,-1,2)$. For example, $$[T]=begin{bmatrix}2&1&2\1&-1&1\1&2&1end{bmatrix}implies T(x,y,z)=(2x+y+2z,x-y+z,x+2y+z)$$
There are infinitely many answers.
answered Jan 4 at 16:55
Shubham JohriShubham Johri
4,524717
answered Jan 4 at 16:55
Shubham JohriShubham Johri
4,524717
answered Jan 4 at 16:55
Shubham JohriShubham Johri
4,524717
answered Jan 4 at 16:55
Shubham JohriShubham Johri
4,524717
4,524717
add a comment |
add a comment |
3
The codomain is four-dimensional, and therefore the image must be generated by four-dimensional vectors.
– Arthur
Jan 4 at 14:13
I do not understand difference between linear operator and linear transformation. So I'm just asking for help.
– Lidy Monteiro
Jan 4 at 14:26
There is no difference. Linear map is also sometimes used. Means the same thing.
– Arthur
Jan 4 at 14:28