How can a matrix act on a set
Question from a PhD entrance exam
If $A=$ begin{bmatrix} 2&-1\-1&2 end{bmatrix}
and $X={xin Bbb R^2:|x|<1}$ where $|x|=|x_1|+|x_2|$
Find $AX$.
Now I know that how $|x|<1$ looks like.It looks like https://i.stack.imgur.com/g09Pd.png
But I dont know what is meant by $AX$ .
How can a matrix act on a set?
Can someone please help me.
EDIT
Using @RobertIsrael
The extreme points are $(1,0),(0,1),(0,-1),(-1,0)$
Now by multiplying $A$ with each one of them we get $A(1,0)=(2,-1)^T,A(0,1)=(-1,2)^T,A(-1,0)=(-2,1)^T,A(0,-1)=(1,-2)^T$
linear-algebra matrices metric-spaces norm
add a comment |
Question from a PhD entrance exam
If $A=$ begin{bmatrix} 2&-1\-1&2 end{bmatrix}
and $X={xin Bbb R^2:|x|<1}$ where $|x|=|x_1|+|x_2|$
Find $AX$.
Now I know that how $|x|<1$ looks like.It looks like https://i.stack.imgur.com/g09Pd.png
But I dont know what is meant by $AX$ .
How can a matrix act on a set?
Can someone please help me.
EDIT
Using @RobertIsrael
The extreme points are $(1,0),(0,1),(0,-1),(-1,0)$
Now by multiplying $A$ with each one of them we get $A(1,0)=(2,-1)^T,A(0,1)=(-1,2)^T,A(-1,0)=(-2,1)^T,A(0,-1)=(1,-2)^T$
linear-algebra matrices metric-spaces norm
2
All the possible vectors $Ax$ where $x$ is in $X$.
– Gordon Royle
Nov 28 '18 at 4:27
It's the image of $X$ under $A$.
– Randall
Nov 28 '18 at 4:28
I would suggest drawing a graph and noting down the transformation of extreme points.
– Thomas Shelby
Nov 28 '18 at 4:33
add a comment |
Question from a PhD entrance exam
If $A=$ begin{bmatrix} 2&-1\-1&2 end{bmatrix}
and $X={xin Bbb R^2:|x|<1}$ where $|x|=|x_1|+|x_2|$
Find $AX$.
Now I know that how $|x|<1$ looks like.It looks like https://i.stack.imgur.com/g09Pd.png
But I dont know what is meant by $AX$ .
How can a matrix act on a set?
Can someone please help me.
EDIT
Using @RobertIsrael
The extreme points are $(1,0),(0,1),(0,-1),(-1,0)$
Now by multiplying $A$ with each one of them we get $A(1,0)=(2,-1)^T,A(0,1)=(-1,2)^T,A(-1,0)=(-2,1)^T,A(0,-1)=(1,-2)^T$
linear-algebra matrices metric-spaces norm
Question from a PhD entrance exam
If $A=$ begin{bmatrix} 2&-1\-1&2 end{bmatrix}
and $X={xin Bbb R^2:|x|<1}$ where $|x|=|x_1|+|x_2|$
Find $AX$.
Now I know that how $|x|<1$ looks like.It looks like https://i.stack.imgur.com/g09Pd.png
But I dont know what is meant by $AX$ .
How can a matrix act on a set?
Can someone please help me.
EDIT
Using @RobertIsrael
The extreme points are $(1,0),(0,1),(0,-1),(-1,0)$
Now by multiplying $A$ with each one of them we get $A(1,0)=(2,-1)^T,A(0,1)=(-1,2)^T,A(-1,0)=(-2,1)^T,A(0,-1)=(1,-2)^T$
linear-algebra matrices metric-spaces norm
linear-algebra matrices metric-spaces norm
edited Nov 28 '18 at 4:37
Join_PhD
asked Nov 28 '18 at 4:25
Join_PhDJoin_PhD
3088
3088
2
All the possible vectors $Ax$ where $x$ is in $X$.
– Gordon Royle
Nov 28 '18 at 4:27
It's the image of $X$ under $A$.
– Randall
Nov 28 '18 at 4:28
I would suggest drawing a graph and noting down the transformation of extreme points.
– Thomas Shelby
Nov 28 '18 at 4:33
add a comment |
2
All the possible vectors $Ax$ where $x$ is in $X$.
– Gordon Royle
Nov 28 '18 at 4:27
It's the image of $X$ under $A$.
– Randall
Nov 28 '18 at 4:28
I would suggest drawing a graph and noting down the transformation of extreme points.
– Thomas Shelby
Nov 28 '18 at 4:33
2
2
All the possible vectors $Ax$ where $x$ is in $X$.
– Gordon Royle
Nov 28 '18 at 4:27
All the possible vectors $Ax$ where $x$ is in $X$.
– Gordon Royle
Nov 28 '18 at 4:27
It's the image of $X$ under $A$.
– Randall
Nov 28 '18 at 4:28
It's the image of $X$ under $A$.
– Randall
Nov 28 '18 at 4:28
I would suggest drawing a graph and noting down the transformation of extreme points.
– Thomas Shelby
Nov 28 '18 at 4:33
I would suggest drawing a graph and noting down the transformation of extreme points.
– Thomas Shelby
Nov 28 '18 at 4:33
add a comment |
1 Answer
1
active
oldest
votes
The notation $AX$ is almost certainly shorthand for the set
$$ {Ax | xin X} $$
which is the image of the set $X$ under the map $A$.
Here are a couple of ideas to get you started on characterizing $AX$. I recommend you work them out on your own --- especially if you plan to be taking a PhD entrance exam any time soon!
- Draw a picture of the action of $A$ on $mathbb{R}^2$. In conjunction with your picture of $X$, this will help you understand what $AX$ will be.
- Characterize $X$ more precisely than simply drawing a picture. One way of doing it is to write elements of $X$ as linear combinations of some set, i.e., specify $v_1,v_2$ such that $X = {c_1v_1 + c_2v_2 | c_1,c_2mbox{ satisfy some constraint}}$.
- Use the fact that $A$ is a linear map to draw conclusions about $AX$ from your characterization of $X$.
But how can I find Ax in this case?How can I multiply each element of the set X with A
– Join_PhD
Nov 28 '18 at 4:28
2
Hint: what are the extreme points?
– Robert Israel
Nov 28 '18 at 4:29
1
And what does $A$ do to them?
– Robert Israel
Nov 28 '18 at 4:31
1
And linear transformations preserve convex hulls.
– Henning Makholm
Nov 28 '18 at 4:35
1
$A(t x + (1-t) y) = t A x + (1-t) A y$.
– Robert Israel
Nov 28 '18 at 15:25
|
show 5 more comments
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1 Answer
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1 Answer
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active
oldest
votes
The notation $AX$ is almost certainly shorthand for the set
$$ {Ax | xin X} $$
which is the image of the set $X$ under the map $A$.
Here are a couple of ideas to get you started on characterizing $AX$. I recommend you work them out on your own --- especially if you plan to be taking a PhD entrance exam any time soon!
- Draw a picture of the action of $A$ on $mathbb{R}^2$. In conjunction with your picture of $X$, this will help you understand what $AX$ will be.
- Characterize $X$ more precisely than simply drawing a picture. One way of doing it is to write elements of $X$ as linear combinations of some set, i.e., specify $v_1,v_2$ such that $X = {c_1v_1 + c_2v_2 | c_1,c_2mbox{ satisfy some constraint}}$.
- Use the fact that $A$ is a linear map to draw conclusions about $AX$ from your characterization of $X$.
But how can I find Ax in this case?How can I multiply each element of the set X with A
– Join_PhD
Nov 28 '18 at 4:28
2
Hint: what are the extreme points?
– Robert Israel
Nov 28 '18 at 4:29
1
And what does $A$ do to them?
– Robert Israel
Nov 28 '18 at 4:31
1
And linear transformations preserve convex hulls.
– Henning Makholm
Nov 28 '18 at 4:35
1
$A(t x + (1-t) y) = t A x + (1-t) A y$.
– Robert Israel
Nov 28 '18 at 15:25
|
show 5 more comments
The notation $AX$ is almost certainly shorthand for the set
$$ {Ax | xin X} $$
which is the image of the set $X$ under the map $A$.
Here are a couple of ideas to get you started on characterizing $AX$. I recommend you work them out on your own --- especially if you plan to be taking a PhD entrance exam any time soon!
- Draw a picture of the action of $A$ on $mathbb{R}^2$. In conjunction with your picture of $X$, this will help you understand what $AX$ will be.
- Characterize $X$ more precisely than simply drawing a picture. One way of doing it is to write elements of $X$ as linear combinations of some set, i.e., specify $v_1,v_2$ such that $X = {c_1v_1 + c_2v_2 | c_1,c_2mbox{ satisfy some constraint}}$.
- Use the fact that $A$ is a linear map to draw conclusions about $AX$ from your characterization of $X$.
But how can I find Ax in this case?How can I multiply each element of the set X with A
– Join_PhD
Nov 28 '18 at 4:28
2
Hint: what are the extreme points?
– Robert Israel
Nov 28 '18 at 4:29
1
And what does $A$ do to them?
– Robert Israel
Nov 28 '18 at 4:31
1
And linear transformations preserve convex hulls.
– Henning Makholm
Nov 28 '18 at 4:35
1
$A(t x + (1-t) y) = t A x + (1-t) A y$.
– Robert Israel
Nov 28 '18 at 15:25
|
show 5 more comments
The notation $AX$ is almost certainly shorthand for the set
$$ {Ax | xin X} $$
which is the image of the set $X$ under the map $A$.
Here are a couple of ideas to get you started on characterizing $AX$. I recommend you work them out on your own --- especially if you plan to be taking a PhD entrance exam any time soon!
- Draw a picture of the action of $A$ on $mathbb{R}^2$. In conjunction with your picture of $X$, this will help you understand what $AX$ will be.
- Characterize $X$ more precisely than simply drawing a picture. One way of doing it is to write elements of $X$ as linear combinations of some set, i.e., specify $v_1,v_2$ such that $X = {c_1v_1 + c_2v_2 | c_1,c_2mbox{ satisfy some constraint}}$.
- Use the fact that $A$ is a linear map to draw conclusions about $AX$ from your characterization of $X$.
The notation $AX$ is almost certainly shorthand for the set
$$ {Ax | xin X} $$
which is the image of the set $X$ under the map $A$.
Here are a couple of ideas to get you started on characterizing $AX$. I recommend you work them out on your own --- especially if you plan to be taking a PhD entrance exam any time soon!
- Draw a picture of the action of $A$ on $mathbb{R}^2$. In conjunction with your picture of $X$, this will help you understand what $AX$ will be.
- Characterize $X$ more precisely than simply drawing a picture. One way of doing it is to write elements of $X$ as linear combinations of some set, i.e., specify $v_1,v_2$ such that $X = {c_1v_1 + c_2v_2 | c_1,c_2mbox{ satisfy some constraint}}$.
- Use the fact that $A$ is a linear map to draw conclusions about $AX$ from your characterization of $X$.
edited Nov 28 '18 at 11:56
answered Nov 28 '18 at 4:27
NealNeal
23.5k23785
23.5k23785
But how can I find Ax in this case?How can I multiply each element of the set X with A
– Join_PhD
Nov 28 '18 at 4:28
2
Hint: what are the extreme points?
– Robert Israel
Nov 28 '18 at 4:29
1
And what does $A$ do to them?
– Robert Israel
Nov 28 '18 at 4:31
1
And linear transformations preserve convex hulls.
– Henning Makholm
Nov 28 '18 at 4:35
1
$A(t x + (1-t) y) = t A x + (1-t) A y$.
– Robert Israel
Nov 28 '18 at 15:25
|
show 5 more comments
But how can I find Ax in this case?How can I multiply each element of the set X with A
– Join_PhD
Nov 28 '18 at 4:28
2
Hint: what are the extreme points?
– Robert Israel
Nov 28 '18 at 4:29
1
And what does $A$ do to them?
– Robert Israel
Nov 28 '18 at 4:31
1
And linear transformations preserve convex hulls.
– Henning Makholm
Nov 28 '18 at 4:35
1
$A(t x + (1-t) y) = t A x + (1-t) A y$.
– Robert Israel
Nov 28 '18 at 15:25
But how can I find Ax in this case?How can I multiply each element of the set X with A
– Join_PhD
Nov 28 '18 at 4:28
But how can I find Ax in this case?How can I multiply each element of the set X with A
– Join_PhD
Nov 28 '18 at 4:28
2
2
Hint: what are the extreme points?
– Robert Israel
Nov 28 '18 at 4:29
Hint: what are the extreme points?
– Robert Israel
Nov 28 '18 at 4:29
1
1
And what does $A$ do to them?
– Robert Israel
Nov 28 '18 at 4:31
And what does $A$ do to them?
– Robert Israel
Nov 28 '18 at 4:31
1
1
And linear transformations preserve convex hulls.
– Henning Makholm
Nov 28 '18 at 4:35
And linear transformations preserve convex hulls.
– Henning Makholm
Nov 28 '18 at 4:35
1
1
$A(t x + (1-t) y) = t A x + (1-t) A y$.
– Robert Israel
Nov 28 '18 at 15:25
$A(t x + (1-t) y) = t A x + (1-t) A y$.
– Robert Israel
Nov 28 '18 at 15:25
|
show 5 more comments
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2
All the possible vectors $Ax$ where $x$ is in $X$.
– Gordon Royle
Nov 28 '18 at 4:27
It's the image of $X$ under $A$.
– Randall
Nov 28 '18 at 4:28
I would suggest drawing a graph and noting down the transformation of extreme points.
– Thomas Shelby
Nov 28 '18 at 4:33