Which of the following are vector subspaces of $mathbb{R}^2$?
up vote
-2
down vote
favorite
Which of the following are vector subspaces of $mathbb{R}^2$?
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x ge 0right}$
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x ge 0, y ge 0right}$
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x = 0right}$
I know that $W$ is a subspace of a vector space $V$ if $W$ is a vector space under the operations of addition and scalar multiplication defined in $V$. So what do I do? In each example shall I choose two matrices for example in each one that are in $W$ and check if their sum is in $mathbb{R}^2$? And take a particular scalar and prove that scalar multiplication is also in $mathbb{R}^2$? But what exactly is $mathbb{R}^2$? I read in the book that is the set of all ordered pairs. How exactly does that work with matrices?
Explain each one please!
Thanks for the help!
linear-algebra matrices vector-spaces
edited Nov 21 at 6:20
Monstrous Moonshiner
2,25911337
asked Nov 21 at 4:37
gi2302
103
New contributor
gi2302 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
|
show 3 more comments
up vote
-2
down vote
favorite
Which of the following are vector subspaces of $mathbb{R}^2$?
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x ge 0right}$
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x ge 0, y ge 0right}$
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x = 0right}$
I know that $W$ is a subspace of a vector space $V$ if $W$ is a vector space under the operations of addition and scalar multiplication defined in $V$. So what do I do? In each example shall I choose two matrices for example in each one that are in $W$ and check if their sum is in $mathbb{R}^2$? And take a particular scalar and prove that scalar multiplication is also in $mathbb{R}^2$? But what exactly is $mathbb{R}^2$? I read in the book that is the set of all ordered pairs. How exactly does that work with matrices?
Explain each one please!
Thanks for the help!
linear-algebra matrices vector-spaces
edited Nov 21 at 6:20
Monstrous Moonshiner
2,25911337
asked Nov 21 at 4:37
gi2302
103
New contributor
gi2302 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
Seems like you are a new user . Please show your work . What do you know about subspaces ?.
– Chinmaya mishra
Nov 21 at 4:40
I've taken the liberty of posting the question in the post itself, since linked questions are frowned upon here, but really you should be doing the work of typing out these questions yourself. Questions which give the impression that you want everyone to do your work for you tend to be downvoted, especially if they appear to be homework questions.
– Monstrous Moonshiner
Nov 21 at 4:54
To counteract such impressions, you should mention what sort of things you have tried, or at least give evidence of some sort of personal investment in trying to find the answer to your question.
– Monstrous Moonshiner
Nov 21 at 4:55
Yes. I'm new born in here. Don't really know how things work here. I know that W is a subspace of a vector space V if W is a vector space under the operations of addition and scalar multiplication defined in V. So what do I do? in each matrix shall choose two matrices for example in each one that are in W and give them value to the variables that meet the requirement and check if the addition is in R^2? and take a particular scalar a prove that scalar multiplicacion is also in R^2? But what exactly is R^2? I read in the book that is the set of all ordered pairs.
– gi2302
Nov 21 at 4:59
didn't post the question itself because I'm not used to the math programming for this... if there any easy way to post here the math stuff will appreciate any recommendation.
– gi2302
Nov 21 at 5:03
|
show 3 more comments
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
Which of the following are vector subspaces of $mathbb{R}^2$?
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x ge 0right}$
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x ge 0, y ge 0right}$
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x = 0right}$
I know that $W$ is a subspace of a vector space $V$ if $W$ is a vector space under the operations of addition and scalar multiplication defined in $V$. So what do I do? In each example shall I choose two matrices for example in each one that are in $W$ and check if their sum is in $mathbb{R}^2$? And take a particular scalar and prove that scalar multiplication is also in $mathbb{R}^2$? But what exactly is $mathbb{R}^2$? I read in the book that is the set of all ordered pairs. How exactly does that work with matrices?
Explain each one please!
Thanks for the help!
linear-algebra matrices vector-spaces
edited Nov 21 at 6:20
Monstrous Moonshiner
2,25911337
asked Nov 21 at 4:37
gi2302
103
New contributor
gi2302 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Which of the following are vector subspaces of $mathbb{R}^2$?
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x ge 0right}$
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x ge 0, y ge 0right}$
- $W = left{ begin{pmatrix} x \ y end{pmatrix}, middle|, x = 0right}$
I know that $W$ is a subspace of a vector space $V$ if $W$ is a vector space under the operations of addition and scalar multiplication defined in $V$. So what do I do? In each example shall I choose two matrices for example in each one that are in $W$ and check if their sum is in $mathbb{R}^2$? And take a particular scalar and prove that scalar multiplication is also in $mathbb{R}^2$? But what exactly is $mathbb{R}^2$? I read in the book that is the set of all ordered pairs. How exactly does that work with matrices?
Explain each one please!
Thanks for the help!
linear-algebra matrices vector-spaces
linear-algebra matrices vector-spaces
edited Nov 21 at 6:20
Monstrous Moonshiner
2,25911337
asked Nov 21 at 4:37
gi2302
103
New contributor
gi2302 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Nov 21 at 6:20
Monstrous Moonshiner
2,25911337
asked Nov 21 at 4:37
gi2302
103
New contributor
gi2302 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Nov 21 at 6:20
Monstrous Moonshiner
2,25911337
edited Nov 21 at 6:20
Monstrous Moonshiner
2,25911337
edited Nov 21 at 6:20
Monstrous Moonshiner
2,25911337
2,25911337
asked Nov 21 at 4:37
gi2302
103
New contributor
gi2302 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 21 at 4:37
gi2302
103
asked Nov 21 at 4:37
gi2302
103
103
New contributor
gi2302 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
gi2302 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
gi2302 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
Seems like you are a new user . Please show your work . What do you know about subspaces ?.
– Chinmaya mishra
Nov 21 at 4:40
I've taken the liberty of posting the question in the post itself, since linked questions are frowned upon here, but really you should be doing the work of typing out these questions yourself. Questions which give the impression that you want everyone to do your work for you tend to be downvoted, especially if they appear to be homework questions.
– Monstrous Moonshiner
Nov 21 at 4:54
To counteract such impressions, you should mention what sort of things you have tried, or at least give evidence of some sort of personal investment in trying to find the answer to your question.
– Monstrous Moonshiner
Nov 21 at 4:55
Yes. I'm new born in here. Don't really know how things work here. I know that W is a subspace of a vector space V if W is a vector space under the operations of addition and scalar multiplication defined in V. So what do I do? in each matrix shall choose two matrices for example in each one that are in W and give them value to the variables that meet the requirement and check if the addition is in R^2? and take a particular scalar a prove that scalar multiplicacion is also in R^2? But what exactly is R^2? I read in the book that is the set of all ordered pairs.
– gi2302
Nov 21 at 4:59
didn't post the question itself because I'm not used to the math programming for this... if there any easy way to post here the math stuff will appreciate any recommendation.
– gi2302
Nov 21 at 5:03
|
show 3 more comments
1
Seems like you are a new user . Please show your work . What do you know about subspaces ?.
– Chinmaya mishra
Nov 21 at 4:40
I've taken the liberty of posting the question in the post itself, since linked questions are frowned upon here, but really you should be doing the work of typing out these questions yourself. Questions which give the impression that you want everyone to do your work for you tend to be downvoted, especially if they appear to be homework questions.
– Monstrous Moonshiner
Nov 21 at 4:54
To counteract such impressions, you should mention what sort of things you have tried, or at least give evidence of some sort of personal investment in trying to find the answer to your question.
– Monstrous Moonshiner
Nov 21 at 4:55
Yes. I'm new born in here. Don't really know how things work here. I know that W is a subspace of a vector space V if W is a vector space under the operations of addition and scalar multiplication defined in V. So what do I do? in each matrix shall choose two matrices for example in each one that are in W and give them value to the variables that meet the requirement and check if the addition is in R^2? and take a particular scalar a prove that scalar multiplicacion is also in R^2? But what exactly is R^2? I read in the book that is the set of all ordered pairs.
– gi2302
Nov 21 at 4:59
didn't post the question itself because I'm not used to the math programming for this... if there any easy way to post here the math stuff will appreciate any recommendation.
– gi2302
Nov 21 at 5:03
1
1
Seems like you are a new user . Please show your work . What do you know about subspaces ?.
– Chinmaya mishra
Nov 21 at 4:40
Seems like you are a new user . Please show your work . What do you know about subspaces ?.
– Chinmaya mishra
Nov 21 at 4:40
I've taken the liberty of posting the question in the post itself, since linked questions are frowned upon here, but really you should be doing the work of typing out these questions yourself. Questions which give the impression that you want everyone to do your work for you tend to be downvoted, especially if they appear to be homework questions.
– Monstrous Moonshiner
Nov 21 at 4:54
I've taken the liberty of posting the question in the post itself, since linked questions are frowned upon here, but really you should be doing the work of typing out these questions yourself. Questions which give the impression that you want everyone to do your work for you tend to be downvoted, especially if they appear to be homework questions.
– Monstrous Moonshiner
Nov 21 at 4:54
To counteract such impressions, you should mention what sort of things you have tried, or at least give evidence of some sort of personal investment in trying to find the answer to your question.
– Monstrous Moonshiner
Nov 21 at 4:55
To counteract such impressions, you should mention what sort of things you have tried, or at least give evidence of some sort of personal investment in trying to find the answer to your question.
– Monstrous Moonshiner
Nov 21 at 4:55
Yes. I'm new born in here. Don't really know how things work here. I know that W is a subspace of a vector space V if W is a vector space under the operations of addition and scalar multiplication defined in V. So what do I do? in each matrix shall choose two matrices for example in each one that are in W and give them value to the variables that meet the requirement and check if the addition is in R^2? and take a particular scalar a prove that scalar multiplicacion is also in R^2? But what exactly is R^2? I read in the book that is the set of all ordered pairs.
– gi2302
Nov 21 at 4:59
Yes. I'm new born in here. Don't really know how things work here. I know that W is a subspace of a vector space V if W is a vector space under the operations of addition and scalar multiplication defined in V. So what do I do? in each matrix shall choose two matrices for example in each one that are in W and give them value to the variables that meet the requirement and check if the addition is in R^2? and take a particular scalar a prove that scalar multiplicacion is also in R^2? But what exactly is R^2? I read in the book that is the set of all ordered pairs.
– gi2302
Nov 21 at 4:59
didn't post the question itself because I'm not used to the math programming for this... if there any easy way to post here the math stuff will appreciate any recommendation.
– gi2302
Nov 21 at 5:03
didn't post the question itself because I'm not used to the math programming for this... if there any easy way to post here the math stuff will appreciate any recommendation.
– gi2302
Nov 21 at 5:03
|
show 3 more comments
1 Answer
1
active
oldest
votes
up vote
0
down vote
The first two are not vector subspaces. Consider the vector $(1,1)$. It is contained in each of the two sets, but its additive inverse is not, so neither set is a subspace. The third set is a subspace. Given two vectors in $W$, they are of the form $(0,y_1)$, $(0,y_2)$ and their sum is $(0,y_1 + y_2)$ which is again in $W$, and given any scalar $r$ it follows that $r(0,y_1) = (0,ry_1)$ is again in $W$. Therefore $W$ is closed under addition and scalar multiplication, so it is a vector subspace.
answered Nov 21 at 5:09
Monstrous Moonshiner
2,25911337
can I choose values for y in a) and c) even I don't have information of y?
– gi2302
Nov 21 at 5:33
Yes, there are no restrictions on what values $y$ can be, so we are free to make them whatever we like.
– Monstrous Moonshiner
Nov 21 at 5:34
ok.. and those are in W.. how do I show they are in $mathbb{R}^2$?
– gi2302
Nov 21 at 5:35
btw thanks for the help and the advices...
– gi2302
Nov 21 at 5:37
$W$ by definition is a subset of $mathbb{R}^2$. You seem to be confused by the fact that elements in $mathbb{R}^2$ are being treated both as ordered pairs and as column vectors. As you read in your book, $mathbb{R}^2$ is defined to be the set of ordered pairs of elements of $mathbb{R}$. But for some purposes, it can be convenient to express elements of $mathbb{R}^2$ as column vectors, with the first component at the top and the second component at the bottom. You should regard this as merely a change in notation, they are essentially the same thing...
– Monstrous Moonshiner
Nov 21 at 5:39
|
show 1 more comment
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The first two are not vector subspaces. Consider the vector $(1,1)$. It is contained in each of the two sets, but its additive inverse is not, so neither set is a subspace. The third set is a subspace. Given two vectors in $W$, they are of the form $(0,y_1)$, $(0,y_2)$ and their sum is $(0,y_1 + y_2)$ which is again in $W$, and given any scalar $r$ it follows that $r(0,y_1) = (0,ry_1)$ is again in $W$. Therefore $W$ is closed under addition and scalar multiplication, so it is a vector subspace.
answered Nov 21 at 5:09
Monstrous Moonshiner
2,25911337
can I choose values for y in a) and c) even I don't have information of y?
– gi2302
Nov 21 at 5:33
Yes, there are no restrictions on what values $y$ can be, so we are free to make them whatever we like.
– Monstrous Moonshiner
Nov 21 at 5:34
ok.. and those are in W.. how do I show they are in $mathbb{R}^2$?
– gi2302
Nov 21 at 5:35
btw thanks for the help and the advices...
– gi2302
Nov 21 at 5:37
$W$ by definition is a subset of $mathbb{R}^2$. You seem to be confused by the fact that elements in $mathbb{R}^2$ are being treated both as ordered pairs and as column vectors. As you read in your book, $mathbb{R}^2$ is defined to be the set of ordered pairs of elements of $mathbb{R}$. But for some purposes, it can be convenient to express elements of $mathbb{R}^2$ as column vectors, with the first component at the top and the second component at the bottom. You should regard this as merely a change in notation, they are essentially the same thing...
– Monstrous Moonshiner
Nov 21 at 5:39
|
show 1 more comment
up vote
0
down vote
The first two are not vector subspaces. Consider the vector $(1,1)$. It is contained in each of the two sets, but its additive inverse is not, so neither set is a subspace. The third set is a subspace. Given two vectors in $W$, they are of the form $(0,y_1)$, $(0,y_2)$ and their sum is $(0,y_1 + y_2)$ which is again in $W$, and given any scalar $r$ it follows that $r(0,y_1) = (0,ry_1)$ is again in $W$. Therefore $W$ is closed under addition and scalar multiplication, so it is a vector subspace.
answered Nov 21 at 5:09
Monstrous Moonshiner
2,25911337
can I choose values for y in a) and c) even I don't have information of y?
– gi2302
Nov 21 at 5:33
Yes, there are no restrictions on what values $y$ can be, so we are free to make them whatever we like.
– Monstrous Moonshiner
Nov 21 at 5:34
ok.. and those are in W.. how do I show they are in $mathbb{R}^2$?
– gi2302
Nov 21 at 5:35
btw thanks for the help and the advices...
– gi2302
Nov 21 at 5:37
$W$ by definition is a subset of $mathbb{R}^2$. You seem to be confused by the fact that elements in $mathbb{R}^2$ are being treated both as ordered pairs and as column vectors. As you read in your book, $mathbb{R}^2$ is defined to be the set of ordered pairs of elements of $mathbb{R}$. But for some purposes, it can be convenient to express elements of $mathbb{R}^2$ as column vectors, with the first component at the top and the second component at the bottom. You should regard this as merely a change in notation, they are essentially the same thing...
– Monstrous Moonshiner
Nov 21 at 5:39
|
show 1 more comment
up vote
0
down vote
up vote
0
down vote
The first two are not vector subspaces. Consider the vector $(1,1)$. It is contained in each of the two sets, but its additive inverse is not, so neither set is a subspace. The third set is a subspace. Given two vectors in $W$, they are of the form $(0,y_1)$, $(0,y_2)$ and their sum is $(0,y_1 + y_2)$ which is again in $W$, and given any scalar $r$ it follows that $r(0,y_1) = (0,ry_1)$ is again in $W$. Therefore $W$ is closed under addition and scalar multiplication, so it is a vector subspace.
answered Nov 21 at 5:09
Monstrous Moonshiner
2,25911337
The first two are not vector subspaces. Consider the vector $(1,1)$. It is contained in each of the two sets, but its additive inverse is not, so neither set is a subspace. The third set is a subspace. Given two vectors in $W$, they are of the form $(0,y_1)$, $(0,y_2)$ and their sum is $(0,y_1 + y_2)$ which is again in $W$, and given any scalar $r$ it follows that $r(0,y_1) = (0,ry_1)$ is again in $W$. Therefore $W$ is closed under addition and scalar multiplication, so it is a vector subspace.
answered Nov 21 at 5:09
Monstrous Moonshiner
2,25911337
answered Nov 21 at 5:09
Monstrous Moonshiner
2,25911337
answered Nov 21 at 5:09
Monstrous Moonshiner
2,25911337
answered Nov 21 at 5:09
Monstrous Moonshiner
2,25911337
2,25911337
can I choose values for y in a) and c) even I don't have information of y?
– gi2302
Nov 21 at 5:33
Yes, there are no restrictions on what values $y$ can be, so we are free to make them whatever we like.
– Monstrous Moonshiner
Nov 21 at 5:34
ok.. and those are in W.. how do I show they are in $mathbb{R}^2$?
– gi2302
Nov 21 at 5:35
btw thanks for the help and the advices...
– gi2302
Nov 21 at 5:37
$W$ by definition is a subset of $mathbb{R}^2$. You seem to be confused by the fact that elements in $mathbb{R}^2$ are being treated both as ordered pairs and as column vectors. As you read in your book, $mathbb{R}^2$ is defined to be the set of ordered pairs of elements of $mathbb{R}$. But for some purposes, it can be convenient to express elements of $mathbb{R}^2$ as column vectors, with the first component at the top and the second component at the bottom. You should regard this as merely a change in notation, they are essentially the same thing...
– Monstrous Moonshiner
Nov 21 at 5:39
|
show 1 more comment
can I choose values for y in a) and c) even I don't have information of y?
– gi2302
Nov 21 at 5:33
Yes, there are no restrictions on what values $y$ can be, so we are free to make them whatever we like.
– Monstrous Moonshiner
Nov 21 at 5:34
ok.. and those are in W.. how do I show they are in $mathbb{R}^2$?
– gi2302
Nov 21 at 5:35
btw thanks for the help and the advices...
– gi2302
Nov 21 at 5:37
$W$ by definition is a subset of $mathbb{R}^2$. You seem to be confused by the fact that elements in $mathbb{R}^2$ are being treated both as ordered pairs and as column vectors. As you read in your book, $mathbb{R}^2$ is defined to be the set of ordered pairs of elements of $mathbb{R}$. But for some purposes, it can be convenient to express elements of $mathbb{R}^2$ as column vectors, with the first component at the top and the second component at the bottom. You should regard this as merely a change in notation, they are essentially the same thing...
– Monstrous Moonshiner
Nov 21 at 5:39
can I choose values for y in a) and c) even I don't have information of y?
– gi2302
Nov 21 at 5:33
can I choose values for y in a) and c) even I don't have information of y?
– gi2302
Nov 21 at 5:33
Yes, there are no restrictions on what values $y$ can be, so we are free to make them whatever we like.
– Monstrous Moonshiner
Nov 21 at 5:34
Yes, there are no restrictions on what values $y$ can be, so we are free to make them whatever we like.
– Monstrous Moonshiner
Nov 21 at 5:34
ok.. and those are in W.. how do I show they are in $mathbb{R}^2$?
– gi2302
Nov 21 at 5:35
ok.. and those are in W.. how do I show they are in $mathbb{R}^2$?
– gi2302
Nov 21 at 5:35
btw thanks for the help and the advices...
– gi2302
Nov 21 at 5:37
btw thanks for the help and the advices...
– gi2302
Nov 21 at 5:37
$W$ by definition is a subset of $mathbb{R}^2$. You seem to be confused by the fact that elements in $mathbb{R}^2$ are being treated both as ordered pairs and as column vectors. As you read in your book, $mathbb{R}^2$ is defined to be the set of ordered pairs of elements of $mathbb{R}$. But for some purposes, it can be convenient to express elements of $mathbb{R}^2$ as column vectors, with the first component at the top and the second component at the bottom. You should regard this as merely a change in notation, they are essentially the same thing...
– Monstrous Moonshiner
Nov 21 at 5:39
$W$ by definition is a subset of $mathbb{R}^2$. You seem to be confused by the fact that elements in $mathbb{R}^2$ are being treated both as ordered pairs and as column vectors. As you read in your book, $mathbb{R}^2$ is defined to be the set of ordered pairs of elements of $mathbb{R}$. But for some purposes, it can be convenient to express elements of $mathbb{R}^2$ as column vectors, with the first component at the top and the second component at the bottom. You should regard this as merely a change in notation, they are essentially the same thing...
– Monstrous Moonshiner
Nov 21 at 5:39
|
show 1 more comment
gi2302 is a new contributor. Be nice, and check out our Code of Conduct.
gi2302 is a new contributor. Be nice, and check out our Code of Conduct.
gi2302 is a new contributor. Be nice, and check out our Code of Conduct.
gi2302 is a new contributor. Be nice, and check out our Code of Conduct.
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1
Seems like you are a new user . Please show your work . What do you know about subspaces ?.
– Chinmaya mishra
Nov 21 at 4:40
I've taken the liberty of posting the question in the post itself, since linked questions are frowned upon here, but really you should be doing the work of typing out these questions yourself. Questions which give the impression that you want everyone to do your work for you tend to be downvoted, especially if they appear to be homework questions.
– Monstrous Moonshiner
Nov 21 at 4:54
To counteract such impressions, you should mention what sort of things you have tried, or at least give evidence of some sort of personal investment in trying to find the answer to your question.
– Monstrous Moonshiner
Nov 21 at 4:55
Yes. I'm new born in here. Don't really know how things work here. I know that W is a subspace of a vector space V if W is a vector space under the operations of addition and scalar multiplication defined in V. So what do I do? in each matrix shall choose two matrices for example in each one that are in W and give them value to the variables that meet the requirement and check if the addition is in R^2? and take a particular scalar a prove that scalar multiplicacion is also in R^2? But what exactly is R^2? I read in the book that is the set of all ordered pairs.
– gi2302
Nov 21 at 4:59
didn't post the question itself because I'm not used to the math programming for this... if there any easy way to post here the math stuff will appreciate any recommendation.
– gi2302
Nov 21 at 5:03