Concept of linear mappings are confusing meChange of Basis ConfusionProve that a linear map for complex polynomials is diagonalizableEigenvalues of three given linear operatorsTransforming coordinate system vs objectsCan an $ntimes n$ matrix be reduced to a smaller matrix in any sense?Overview of Linear AlgebraLinear Transformation vs MatrixPruning SubsetsChange of basis formula - intuition/is this true?Linear Algebra:Vector Space
Why don't electron-positron collisions release infinite energy?
TGV timetables / schedules?
Modification to Chariots for Heavy Cavalry Analogue for 4-armed race
How is the claim "I am in New York only if I am in America" the same as "If I am in New York, then I am in America?
How is this relation reflexive?
I probably found a bug with the sudo apt install function
Circuitry of TV splitters
Do airline pilots ever risk not hearing communication directed to them specifically, from traffic controllers?
How can the DM most effectively choose 1 out of an odd number of players to be targeted by an attack or effect?
Why was the small council so happy for Tyrion to become the Master of Coin?
What do you call something that goes against the spirit of the law, but is legal when interpreting the law to the letter?
How old can references or sources in a thesis be?
A function which translates a sentence to title-case
Are tax years 2016 & 2017 back taxes deductible for tax year 2018?
How to add power-LED to my small amplifier?
A newer friend of my brother's gave him a load of baseball cards that are supposedly extremely valuable. Is this a scam?
Is there really no realistic way for a skeleton monster to move around without magic?
What typically incentivizes a professor to change jobs to a lower ranking university?
How do we improve the relationship with a client software team that performs poorly and is becoming less collaborative?
XeLaTeX and pdfLaTeX ignore hyphenation
Is there a minimum number of transactions in a block?
How does one intimidate enemies without having the capacity for violence?
Motorized valve interfering with button?
Extreme, but not acceptable situation and I can't start the work tomorrow morning
Concept of linear mappings are confusing me
Change of Basis ConfusionProve that a linear map for complex polynomials is diagonalizableEigenvalues of three given linear operatorsTransforming coordinate system vs objectsCan an $ntimes n$ matrix be reduced to a smaller matrix in any sense?Overview of Linear AlgebraLinear Transformation vs MatrixPruning SubsetsChange of basis formula - intuition/is this true?Linear Algebra:Vector Space
$begingroup$
I'm so confused on how we can have a 2x3 matrix A, multiply it by a vector in $Bbb R^3$ and then end up with a vector in $Bbb R^2$. Is it possible to visualize this at all or do I need to sort of blindly accept this concept as facts that I'll accept and use?
Can someone give a very brief summarization on why this makes sense? Because I just see it as, in a world (dimension) in $Bbb R^3$, we multiply it by a vector in $Bbb R^3$, and out pops a vector in $Bbb R^2$.
Thanks!
linear-algebra
asked 57 mins ago
mingming
4306
$endgroup$
add a comment |
$begingroup$
I'm so confused on how we can have a 2x3 matrix A, multiply it by a vector in $Bbb R^3$ and then end up with a vector in $Bbb R^2$. Is it possible to visualize this at all or do I need to sort of blindly accept this concept as facts that I'll accept and use?
Can someone give a very brief summarization on why this makes sense? Because I just see it as, in a world (dimension) in $Bbb R^3$, we multiply it by a vector in $Bbb R^3$, and out pops a vector in $Bbb R^2$.
Thanks!
linear-algebra
asked 57 mins ago
mingming
4306
$endgroup$
1
$begingroup$
maybe think of multiplying a matrix by a vector as a special case of multiplying a matrix by a matrix
$endgroup$
– J. W. Tanner
53 mins ago
$begingroup$
Is it the definition of matrix multiplication that gives you trouble? Have you tried doing a multiplication and seeing what you get? Do you understand that we can have a function like $f(x,y,z)=(x-2y+z, 2x+4y-z)$ which maps $mathbb R^3$ to $mathbb R^2$?
$endgroup$
– John Douma
41 mins ago
add a comment |
$begingroup$
I'm so confused on how we can have a 2x3 matrix A, multiply it by a vector in $Bbb R^3$ and then end up with a vector in $Bbb R^2$. Is it possible to visualize this at all or do I need to sort of blindly accept this concept as facts that I'll accept and use?
Can someone give a very brief summarization on why this makes sense? Because I just see it as, in a world (dimension) in $Bbb R^3$, we multiply it by a vector in $Bbb R^3$, and out pops a vector in $Bbb R^2$.
Thanks!
linear-algebra
asked 57 mins ago
mingming
4306
$endgroup$
I'm so confused on how we can have a 2x3 matrix A, multiply it by a vector in $Bbb R^3$ and then end up with a vector in $Bbb R^2$. Is it possible to visualize this at all or do I need to sort of blindly accept this concept as facts that I'll accept and use?
Can someone give a very brief summarization on why this makes sense? Because I just see it as, in a world (dimension) in $Bbb R^3$, we multiply it by a vector in $Bbb R^3$, and out pops a vector in $Bbb R^2$.
Thanks!
linear-algebra
linear-algebra
asked 57 mins ago
mingming
4306
asked 57 mins ago
mingming
4306
asked 57 mins ago
mingming
4306
asked 57 mins ago
mingming
4306
asked 57 mins ago
mingming
4306
4306
1
$begingroup$
maybe think of multiplying a matrix by a vector as a special case of multiplying a matrix by a matrix
$endgroup$
– J. W. Tanner
53 mins ago
$begingroup$
Is it the definition of matrix multiplication that gives you trouble? Have you tried doing a multiplication and seeing what you get? Do you understand that we can have a function like $f(x,y,z)=(x-2y+z, 2x+4y-z)$ which maps $mathbb R^3$ to $mathbb R^2$?
$endgroup$
– John Douma
41 mins ago
add a comment |
1
$begingroup$
maybe think of multiplying a matrix by a vector as a special case of multiplying a matrix by a matrix
$endgroup$
– J. W. Tanner
53 mins ago
$begingroup$
Is it the definition of matrix multiplication that gives you trouble? Have you tried doing a multiplication and seeing what you get? Do you understand that we can have a function like $f(x,y,z)=(x-2y+z, 2x+4y-z)$ which maps $mathbb R^3$ to $mathbb R^2$?
$endgroup$
– John Douma
41 mins ago
1
1
$begingroup$
maybe think of multiplying a matrix by a vector as a special case of multiplying a matrix by a matrix
$endgroup$
– J. W. Tanner
53 mins ago
$begingroup$
maybe think of multiplying a matrix by a vector as a special case of multiplying a matrix by a matrix
$endgroup$
– J. W. Tanner
53 mins ago
$begingroup$
Is it the definition of matrix multiplication that gives you trouble? Have you tried doing a multiplication and seeing what you get? Do you understand that we can have a function like $f(x,y,z)=(x-2y+z, 2x+4y-z)$ which maps $mathbb R^3$ to $mathbb R^2$?
$endgroup$
– John Douma
41 mins ago
$begingroup$
Is it the definition of matrix multiplication that gives you trouble? Have you tried doing a multiplication and seeing what you get? Do you understand that we can have a function like $f(x,y,z)=(x-2y+z, 2x+4y-z)$ which maps $mathbb R^3$ to $mathbb R^2$?
$endgroup$
– John Douma
41 mins ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
For the moment don't think about multiplication and matrices.
You can imagine starting from a vector $(x,y,z)$ in $mathbbR^3$ and mapping it to a vector in $mathbbR^2$ this way, for example:
$$
(x, y, z) mapsto (2x+ z, 3x+ 4y).
$$
Mathematicians have invented a nice clean way to write that map. It's the formalism you've learned for matrix multiplication. To see what $(1,2,3)$ maps to, calculate the matrix product
$$
beginbmatrix
2 & 0 & 1 \
3 & 4 & 0
endbmatrix
beginbmatrix
1 \
2 \
3
endbmatrix
=
beginbmatrix
5\
11
endbmatrix.
$$
You will soon be comfortable with this, just as you are now with whatever algorithm you were taught for ordinary multiplication. Then you will be free to focus on understanding what maps like this are useful for.
answered 42 mins ago
Ethan BolkerEthan Bolker
45.8k553120
$endgroup$
add a comment |
$begingroup$
A linear mapping has the property that it maps subspaces to subspaces.
So it will map a line to a line or $0$, a plane to a plane, a line, or $0$, and so on.
By definition, linear mappings “play nice” with addition and scaling. These properties allow us to reduce statements about entire vector spaces down to bases, which are quite “small” in the finite dimensional case.
answered 32 mins ago
rschwiebrschwieb
108k12103253
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3179032%2fconcept-of-linear-mappings-are-confusing-me%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For the moment don't think about multiplication and matrices.
You can imagine starting from a vector $(x,y,z)$ in $mathbbR^3$ and mapping it to a vector in $mathbbR^2$ this way, for example:
$$
(x, y, z) mapsto (2x+ z, 3x+ 4y).
$$
Mathematicians have invented a nice clean way to write that map. It's the formalism you've learned for matrix multiplication. To see what $(1,2,3)$ maps to, calculate the matrix product
$$
beginbmatrix
2 & 0 & 1 \
3 & 4 & 0
endbmatrix
beginbmatrix
1 \
2 \
3
endbmatrix
=
beginbmatrix
5\
11
endbmatrix.
$$
You will soon be comfortable with this, just as you are now with whatever algorithm you were taught for ordinary multiplication. Then you will be free to focus on understanding what maps like this are useful for.
answered 42 mins ago
Ethan BolkerEthan Bolker
45.8k553120
$endgroup$
add a comment |
$begingroup$
For the moment don't think about multiplication and matrices.
You can imagine starting from a vector $(x,y,z)$ in $mathbbR^3$ and mapping it to a vector in $mathbbR^2$ this way, for example:
$$
(x, y, z) mapsto (2x+ z, 3x+ 4y).
$$
Mathematicians have invented a nice clean way to write that map. It's the formalism you've learned for matrix multiplication. To see what $(1,2,3)$ maps to, calculate the matrix product
$$
beginbmatrix
2 & 0 & 1 \
3 & 4 & 0
endbmatrix
beginbmatrix
1 \
2 \
3
endbmatrix
=
beginbmatrix
5\
11
endbmatrix.
$$
You will soon be comfortable with this, just as you are now with whatever algorithm you were taught for ordinary multiplication. Then you will be free to focus on understanding what maps like this are useful for.
answered 42 mins ago
Ethan BolkerEthan Bolker
45.8k553120
$endgroup$
add a comment |
$begingroup$
For the moment don't think about multiplication and matrices.
You can imagine starting from a vector $(x,y,z)$ in $mathbbR^3$ and mapping it to a vector in $mathbbR^2$ this way, for example:
$$
(x, y, z) mapsto (2x+ z, 3x+ 4y).
$$
Mathematicians have invented a nice clean way to write that map. It's the formalism you've learned for matrix multiplication. To see what $(1,2,3)$ maps to, calculate the matrix product
$$
beginbmatrix
2 & 0 & 1 \
3 & 4 & 0
endbmatrix
beginbmatrix
1 \
2 \
3
endbmatrix
=
beginbmatrix
5\
11
endbmatrix.
$$
You will soon be comfortable with this, just as you are now with whatever algorithm you were taught for ordinary multiplication. Then you will be free to focus on understanding what maps like this are useful for.
answered 42 mins ago
Ethan BolkerEthan Bolker
45.8k553120
$endgroup$
For the moment don't think about multiplication and matrices.
You can imagine starting from a vector $(x,y,z)$ in $mathbbR^3$ and mapping it to a vector in $mathbbR^2$ this way, for example:
$$
(x, y, z) mapsto (2x+ z, 3x+ 4y).
$$
Mathematicians have invented a nice clean way to write that map. It's the formalism you've learned for matrix multiplication. To see what $(1,2,3)$ maps to, calculate the matrix product
$$
beginbmatrix
2 & 0 & 1 \
3 & 4 & 0
endbmatrix
beginbmatrix
1 \
2 \
3
endbmatrix
=
beginbmatrix
5\
11
endbmatrix.
$$
You will soon be comfortable with this, just as you are now with whatever algorithm you were taught for ordinary multiplication. Then you will be free to focus on understanding what maps like this are useful for.
answered 42 mins ago
Ethan BolkerEthan Bolker
45.8k553120
answered 42 mins ago
Ethan BolkerEthan Bolker
45.8k553120
answered 42 mins ago
Ethan BolkerEthan Bolker
45.8k553120
answered 42 mins ago
Ethan BolkerEthan Bolker
45.8k553120
45.8k553120
add a comment |
add a comment |
$begingroup$
A linear mapping has the property that it maps subspaces to subspaces.
So it will map a line to a line or $0$, a plane to a plane, a line, or $0$, and so on.
By definition, linear mappings “play nice” with addition and scaling. These properties allow us to reduce statements about entire vector spaces down to bases, which are quite “small” in the finite dimensional case.
answered 32 mins ago
rschwiebrschwieb
108k12103253
$endgroup$
add a comment |
$begingroup$
A linear mapping has the property that it maps subspaces to subspaces.
So it will map a line to a line or $0$, a plane to a plane, a line, or $0$, and so on.
By definition, linear mappings “play nice” with addition and scaling. These properties allow us to reduce statements about entire vector spaces down to bases, which are quite “small” in the finite dimensional case.
answered 32 mins ago
rschwiebrschwieb
108k12103253
$endgroup$
add a comment |
$begingroup$
A linear mapping has the property that it maps subspaces to subspaces.
So it will map a line to a line or $0$, a plane to a plane, a line, or $0$, and so on.
By definition, linear mappings “play nice” with addition and scaling. These properties allow us to reduce statements about entire vector spaces down to bases, which are quite “small” in the finite dimensional case.
answered 32 mins ago
rschwiebrschwieb
108k12103253
$endgroup$
A linear mapping has the property that it maps subspaces to subspaces.
So it will map a line to a line or $0$, a plane to a plane, a line, or $0$, and so on.
By definition, linear mappings “play nice” with addition and scaling. These properties allow us to reduce statements about entire vector spaces down to bases, which are quite “small” in the finite dimensional case.
answered 32 mins ago
rschwiebrschwieb
108k12103253
answered 32 mins ago
rschwiebrschwieb
108k12103253
answered 32 mins ago
rschwiebrschwieb
108k12103253
answered 32 mins ago
rschwiebrschwieb
108k12103253
108k12103253
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3179032%2fconcept-of-linear-mappings-are-confusing-me%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
maybe think of multiplying a matrix by a vector as a special case of multiplying a matrix by a matrix
$endgroup$
– J. W. Tanner
53 mins ago
$begingroup$
Is it the definition of matrix multiplication that gives you trouble? Have you tried doing a multiplication and seeing what you get? Do you understand that we can have a function like $f(x,y,z)=(x-2y+z, 2x+4y-z)$ which maps $mathbb R^3$ to $mathbb R^2$?
$endgroup$
– John Douma
41 mins ago