How can a matrix act on a set












0














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







share|cite|improve this question




















  • 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
















0














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







share|cite|improve this question




















  • 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














0












0








0


1





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







share|cite|improve this question















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






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share|cite|improve this question













share|cite|improve this question




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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














  • 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










1 Answer
1






active

oldest

votes


















1














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 $X$.






share|cite|improve this answer























  • 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













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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









1














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 $X$.






share|cite|improve this answer























  • 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


















1














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 $X$.






share|cite|improve this answer























  • 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
















1












1








1






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 $X$.






share|cite|improve this answer














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 $X$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








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




















  • 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




















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