Invariants between two isomorphic vector spacesComplex vector spacesSimple questions about isomorphisms between vector spacesWhat is needed to make Euclidean spaces isomorphic as groups?Vector Spaces and GroupsWhat do mathematicians mean by if two vector spaces are isomorphic, then “they are the same”What are the implications of the fact that vector spaces are isomorphic?Assume that $V$ and $W$ are isomorphic vector spaces. What does it imply if $V$ and $W$ are not finite dimensional.If there is a bijective linear transformation between two vector spaces, do they bear same properties?Are $C[0,1]$ and $C[0,1)$ isomorphic?Subspaces and elements of isomorphic vector spaces

Issue with type force PATH search

Personal Teleportation: From Rags to Riches

Why does Optional.map make this assignment work?

Infinite Abelian subgroup of infinite non Abelian group example

What exploit are these user agents trying to use?

Should I tell management that I intend to leave due to bad software development practices?

Why do bosons tend to occupy the same state?

What reasons are there for a Capitalist to oppose a 100% inheritance tax?

Is it possible to download Internet Explorer on my Mac running OS X El Capitan?

Can a rocket refuel on Mars from water?

Where does SFDX store details about scratch orgs?

How seriously should I take size and weight limits of hand luggage?

Watching something be piped to a file live with tail

Why was the shrinking from 8″ made only to 5.25″ and not smaller (4″ or less)?

Magento 2: Migrate only Customer and orders

Alternative to sending password over mail?

Dealing with conflict between co-workers for non-work-related issue affecting their work

How could indestructible materials be used in power generation?

What is the formal way to express the meaning of a variable?

Twin primes whose sum is a cube

How can I prevent hyper evolved versions of regular creatures from wiping out their cousins?

Can I ask the recruiters in my resume to put the reason why I am rejected?

What is the difference between Drupal::request()->getSession() and Drupal::service('user.private_tempstore')?

What is the most common color to indicate the input-field is disabled?



Invariants between two isomorphic vector spaces


Complex vector spacesSimple questions about isomorphisms between vector spacesWhat is needed to make Euclidean spaces isomorphic as groups?Vector Spaces and GroupsWhat do mathematicians mean by if two vector spaces are isomorphic, then “they are the same”What are the implications of the fact that vector spaces are isomorphic?Assume that $V$ and $W$ are isomorphic vector spaces. What does it imply if $V$ and $W$ are not finite dimensional.If there is a bijective linear transformation between two vector spaces, do they bear same properties?Are $C[0,1]$ and $C[0,1)$ isomorphic?Subspaces and elements of isomorphic vector spaces













2












$begingroup$


I have a general question about isomorphisms between vector spaces.



From a general point of view, there are common properties (invariants) between two isomorphic structures (e.g., properties about compactness or connectedness between two isomorphic topological spaces, commutativity between two isomorphic groups). Can you give me some examples of invariants between two isomorphic vector spaces ?



Thank you for your help !










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Dimension certainly.
    $endgroup$
    – Wuestenfux
    Mar 26 at 14:15















2












$begingroup$


I have a general question about isomorphisms between vector spaces.



From a general point of view, there are common properties (invariants) between two isomorphic structures (e.g., properties about compactness or connectedness between two isomorphic topological spaces, commutativity between two isomorphic groups). Can you give me some examples of invariants between two isomorphic vector spaces ?



Thank you for your help !










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Dimension certainly.
    $endgroup$
    – Wuestenfux
    Mar 26 at 14:15













2












2








2





$begingroup$


I have a general question about isomorphisms between vector spaces.



From a general point of view, there are common properties (invariants) between two isomorphic structures (e.g., properties about compactness or connectedness between two isomorphic topological spaces, commutativity between two isomorphic groups). Can you give me some examples of invariants between two isomorphic vector spaces ?



Thank you for your help !










share|cite|improve this question











$endgroup$




I have a general question about isomorphisms between vector spaces.



From a general point of view, there are common properties (invariants) between two isomorphic structures (e.g., properties about compactness or connectedness between two isomorphic topological spaces, commutativity between two isomorphic groups). Can you give me some examples of invariants between two isomorphic vector spaces ?



Thank you for your help !







vector-spaces vector-space-isomorphism invariance






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 26 at 15:01









José Carlos Santos

172k22132239




172k22132239










asked Mar 26 at 14:13









deeppinkwaterdeeppinkwater

858




858







  • 1




    $begingroup$
    Dimension certainly.
    $endgroup$
    – Wuestenfux
    Mar 26 at 14:15












  • 1




    $begingroup$
    Dimension certainly.
    $endgroup$
    – Wuestenfux
    Mar 26 at 14:15







1




1




$begingroup$
Dimension certainly.
$endgroup$
– Wuestenfux
Mar 26 at 14:15




$begingroup$
Dimension certainly.
$endgroup$
– Wuestenfux
Mar 26 at 14:15










2 Answers
2






active

oldest

votes


















6












$begingroup$

One of the fundamental theorems of linear algebra is that there is only one invariant you really need to care about: the dimension. Two vector spaces are isomorphic if and only if they have the same dimension. The reason is that any map that bijectively takes a basis of one vector space to a basis of another vector space must extend linearly (in a unique way) to a vector space isomorphism.



Another example is the cardinality of the underlying set (of course, the isomorphism is a bijection).



This observation is actually useful in field theory. Suppose you have a field $K$ which you know is finite. Then the prime subfield of $K$ is $mathbbF_p$ for some prime $p.$ Now $K$ must have finite dimension as a vector space over $mathbbF_p$ (since $K$ is finite), so, again as a vector space, $K=(mathbbF_p)^oplus r$ for some $r.$ It follows immediately that $lvertKrvert=p^r$ for the same $r.$ Moreover, if two finite fields are of the same cardinality, then that cardinality must be $p^r$ for some prime $p$ and some integer $r,$ and it follows that those two fields are isomorphic as vector spaces over the prime subfield $mathbbF_p.$ (With a bit more work, one can show that any two finite fields of the same cardinality are isomorphic as fields, and that a finite field of cardinality $p^r$ does exist for every prime $p$ and every integer $r>0.$)






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Thank you for this answer.
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:54










  • $begingroup$
    @deeppinkwater: It is perhaps also worth noting that although (or rather, because!) the dimension is the only thing that determines the isomorphism class of a vector space, most linear algebra courses actually spend more time focusing on the linear maps between vector spaces. Then there are lots of "invariants", depending upon your notion of isomorphism, the primary examples being eigenvalues (with their respective multiplicities), and therefore also the determinant and the trace, and the singular values.
    $endgroup$
    – Will R
    Mar 26 at 17:13











  • $begingroup$
    I take it by "underlying set", you mean "scalar field"? I think that the latter is more precise.
    $endgroup$
    – Acccumulation
    Mar 26 at 21:39










  • $begingroup$
    @Acccumulation: No, I mean the set of vectors. If a vector space over a field $K$ is an abelian group $V$ with a linear action of $K$ on $V$ (basically, you have some operations and a load of axioms are satisfied), then by "underlying set" I mean the set $V.$
    $endgroup$
    – Will R
    Mar 26 at 21:41


















2












$begingroup$

The number one is the dimension, of course, since two vector spaces over the same field are isomorphic if and only if they have the same dimension.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thank you, that's the only one that came to my mind actually !
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:16











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
);



);













draft saved

draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3163230%2finvariants-between-two-isomorphic-vector-spaces%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









6












$begingroup$

One of the fundamental theorems of linear algebra is that there is only one invariant you really need to care about: the dimension. Two vector spaces are isomorphic if and only if they have the same dimension. The reason is that any map that bijectively takes a basis of one vector space to a basis of another vector space must extend linearly (in a unique way) to a vector space isomorphism.



Another example is the cardinality of the underlying set (of course, the isomorphism is a bijection).



This observation is actually useful in field theory. Suppose you have a field $K$ which you know is finite. Then the prime subfield of $K$ is $mathbbF_p$ for some prime $p.$ Now $K$ must have finite dimension as a vector space over $mathbbF_p$ (since $K$ is finite), so, again as a vector space, $K=(mathbbF_p)^oplus r$ for some $r.$ It follows immediately that $lvertKrvert=p^r$ for the same $r.$ Moreover, if two finite fields are of the same cardinality, then that cardinality must be $p^r$ for some prime $p$ and some integer $r,$ and it follows that those two fields are isomorphic as vector spaces over the prime subfield $mathbbF_p.$ (With a bit more work, one can show that any two finite fields of the same cardinality are isomorphic as fields, and that a finite field of cardinality $p^r$ does exist for every prime $p$ and every integer $r>0.$)






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Thank you for this answer.
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:54










  • $begingroup$
    @deeppinkwater: It is perhaps also worth noting that although (or rather, because!) the dimension is the only thing that determines the isomorphism class of a vector space, most linear algebra courses actually spend more time focusing on the linear maps between vector spaces. Then there are lots of "invariants", depending upon your notion of isomorphism, the primary examples being eigenvalues (with their respective multiplicities), and therefore also the determinant and the trace, and the singular values.
    $endgroup$
    – Will R
    Mar 26 at 17:13











  • $begingroup$
    I take it by "underlying set", you mean "scalar field"? I think that the latter is more precise.
    $endgroup$
    – Acccumulation
    Mar 26 at 21:39










  • $begingroup$
    @Acccumulation: No, I mean the set of vectors. If a vector space over a field $K$ is an abelian group $V$ with a linear action of $K$ on $V$ (basically, you have some operations and a load of axioms are satisfied), then by "underlying set" I mean the set $V.$
    $endgroup$
    – Will R
    Mar 26 at 21:41















6












$begingroup$

One of the fundamental theorems of linear algebra is that there is only one invariant you really need to care about: the dimension. Two vector spaces are isomorphic if and only if they have the same dimension. The reason is that any map that bijectively takes a basis of one vector space to a basis of another vector space must extend linearly (in a unique way) to a vector space isomorphism.



Another example is the cardinality of the underlying set (of course, the isomorphism is a bijection).



This observation is actually useful in field theory. Suppose you have a field $K$ which you know is finite. Then the prime subfield of $K$ is $mathbbF_p$ for some prime $p.$ Now $K$ must have finite dimension as a vector space over $mathbbF_p$ (since $K$ is finite), so, again as a vector space, $K=(mathbbF_p)^oplus r$ for some $r.$ It follows immediately that $lvertKrvert=p^r$ for the same $r.$ Moreover, if two finite fields are of the same cardinality, then that cardinality must be $p^r$ for some prime $p$ and some integer $r,$ and it follows that those two fields are isomorphic as vector spaces over the prime subfield $mathbbF_p.$ (With a bit more work, one can show that any two finite fields of the same cardinality are isomorphic as fields, and that a finite field of cardinality $p^r$ does exist for every prime $p$ and every integer $r>0.$)






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Thank you for this answer.
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:54










  • $begingroup$
    @deeppinkwater: It is perhaps also worth noting that although (or rather, because!) the dimension is the only thing that determines the isomorphism class of a vector space, most linear algebra courses actually spend more time focusing on the linear maps between vector spaces. Then there are lots of "invariants", depending upon your notion of isomorphism, the primary examples being eigenvalues (with their respective multiplicities), and therefore also the determinant and the trace, and the singular values.
    $endgroup$
    – Will R
    Mar 26 at 17:13











  • $begingroup$
    I take it by "underlying set", you mean "scalar field"? I think that the latter is more precise.
    $endgroup$
    – Acccumulation
    Mar 26 at 21:39










  • $begingroup$
    @Acccumulation: No, I mean the set of vectors. If a vector space over a field $K$ is an abelian group $V$ with a linear action of $K$ on $V$ (basically, you have some operations and a load of axioms are satisfied), then by "underlying set" I mean the set $V.$
    $endgroup$
    – Will R
    Mar 26 at 21:41













6












6








6





$begingroup$

One of the fundamental theorems of linear algebra is that there is only one invariant you really need to care about: the dimension. Two vector spaces are isomorphic if and only if they have the same dimension. The reason is that any map that bijectively takes a basis of one vector space to a basis of another vector space must extend linearly (in a unique way) to a vector space isomorphism.



Another example is the cardinality of the underlying set (of course, the isomorphism is a bijection).



This observation is actually useful in field theory. Suppose you have a field $K$ which you know is finite. Then the prime subfield of $K$ is $mathbbF_p$ for some prime $p.$ Now $K$ must have finite dimension as a vector space over $mathbbF_p$ (since $K$ is finite), so, again as a vector space, $K=(mathbbF_p)^oplus r$ for some $r.$ It follows immediately that $lvertKrvert=p^r$ for the same $r.$ Moreover, if two finite fields are of the same cardinality, then that cardinality must be $p^r$ for some prime $p$ and some integer $r,$ and it follows that those two fields are isomorphic as vector spaces over the prime subfield $mathbbF_p.$ (With a bit more work, one can show that any two finite fields of the same cardinality are isomorphic as fields, and that a finite field of cardinality $p^r$ does exist for every prime $p$ and every integer $r>0.$)






share|cite|improve this answer











$endgroup$



One of the fundamental theorems of linear algebra is that there is only one invariant you really need to care about: the dimension. Two vector spaces are isomorphic if and only if they have the same dimension. The reason is that any map that bijectively takes a basis of one vector space to a basis of another vector space must extend linearly (in a unique way) to a vector space isomorphism.



Another example is the cardinality of the underlying set (of course, the isomorphism is a bijection).



This observation is actually useful in field theory. Suppose you have a field $K$ which you know is finite. Then the prime subfield of $K$ is $mathbbF_p$ for some prime $p.$ Now $K$ must have finite dimension as a vector space over $mathbbF_p$ (since $K$ is finite), so, again as a vector space, $K=(mathbbF_p)^oplus r$ for some $r.$ It follows immediately that $lvertKrvert=p^r$ for the same $r.$ Moreover, if two finite fields are of the same cardinality, then that cardinality must be $p^r$ for some prime $p$ and some integer $r,$ and it follows that those two fields are isomorphic as vector spaces over the prime subfield $mathbbF_p.$ (With a bit more work, one can show that any two finite fields of the same cardinality are isomorphic as fields, and that a finite field of cardinality $p^r$ does exist for every prime $p$ and every integer $r>0.$)







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Mar 26 at 20:27

























answered Mar 26 at 14:29









Will RWill R

6,78231429




6,78231429











  • $begingroup$
    Thank you for this answer.
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:54










  • $begingroup$
    @deeppinkwater: It is perhaps also worth noting that although (or rather, because!) the dimension is the only thing that determines the isomorphism class of a vector space, most linear algebra courses actually spend more time focusing on the linear maps between vector spaces. Then there are lots of "invariants", depending upon your notion of isomorphism, the primary examples being eigenvalues (with their respective multiplicities), and therefore also the determinant and the trace, and the singular values.
    $endgroup$
    – Will R
    Mar 26 at 17:13











  • $begingroup$
    I take it by "underlying set", you mean "scalar field"? I think that the latter is more precise.
    $endgroup$
    – Acccumulation
    Mar 26 at 21:39










  • $begingroup$
    @Acccumulation: No, I mean the set of vectors. If a vector space over a field $K$ is an abelian group $V$ with a linear action of $K$ on $V$ (basically, you have some operations and a load of axioms are satisfied), then by "underlying set" I mean the set $V.$
    $endgroup$
    – Will R
    Mar 26 at 21:41
















  • $begingroup$
    Thank you for this answer.
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:54










  • $begingroup$
    @deeppinkwater: It is perhaps also worth noting that although (or rather, because!) the dimension is the only thing that determines the isomorphism class of a vector space, most linear algebra courses actually spend more time focusing on the linear maps between vector spaces. Then there are lots of "invariants", depending upon your notion of isomorphism, the primary examples being eigenvalues (with their respective multiplicities), and therefore also the determinant and the trace, and the singular values.
    $endgroup$
    – Will R
    Mar 26 at 17:13











  • $begingroup$
    I take it by "underlying set", you mean "scalar field"? I think that the latter is more precise.
    $endgroup$
    – Acccumulation
    Mar 26 at 21:39










  • $begingroup$
    @Acccumulation: No, I mean the set of vectors. If a vector space over a field $K$ is an abelian group $V$ with a linear action of $K$ on $V$ (basically, you have some operations and a load of axioms are satisfied), then by "underlying set" I mean the set $V.$
    $endgroup$
    – Will R
    Mar 26 at 21:41















$begingroup$
Thank you for this answer.
$endgroup$
– deeppinkwater
Mar 26 at 14:54




$begingroup$
Thank you for this answer.
$endgroup$
– deeppinkwater
Mar 26 at 14:54












$begingroup$
@deeppinkwater: It is perhaps also worth noting that although (or rather, because!) the dimension is the only thing that determines the isomorphism class of a vector space, most linear algebra courses actually spend more time focusing on the linear maps between vector spaces. Then there are lots of "invariants", depending upon your notion of isomorphism, the primary examples being eigenvalues (with their respective multiplicities), and therefore also the determinant and the trace, and the singular values.
$endgroup$
– Will R
Mar 26 at 17:13





$begingroup$
@deeppinkwater: It is perhaps also worth noting that although (or rather, because!) the dimension is the only thing that determines the isomorphism class of a vector space, most linear algebra courses actually spend more time focusing on the linear maps between vector spaces. Then there are lots of "invariants", depending upon your notion of isomorphism, the primary examples being eigenvalues (with their respective multiplicities), and therefore also the determinant and the trace, and the singular values.
$endgroup$
– Will R
Mar 26 at 17:13













$begingroup$
I take it by "underlying set", you mean "scalar field"? I think that the latter is more precise.
$endgroup$
– Acccumulation
Mar 26 at 21:39




$begingroup$
I take it by "underlying set", you mean "scalar field"? I think that the latter is more precise.
$endgroup$
– Acccumulation
Mar 26 at 21:39












$begingroup$
@Acccumulation: No, I mean the set of vectors. If a vector space over a field $K$ is an abelian group $V$ with a linear action of $K$ on $V$ (basically, you have some operations and a load of axioms are satisfied), then by "underlying set" I mean the set $V.$
$endgroup$
– Will R
Mar 26 at 21:41




$begingroup$
@Acccumulation: No, I mean the set of vectors. If a vector space over a field $K$ is an abelian group $V$ with a linear action of $K$ on $V$ (basically, you have some operations and a load of axioms are satisfied), then by "underlying set" I mean the set $V.$
$endgroup$
– Will R
Mar 26 at 21:41











2












$begingroup$

The number one is the dimension, of course, since two vector spaces over the same field are isomorphic if and only if they have the same dimension.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thank you, that's the only one that came to my mind actually !
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:16















2












$begingroup$

The number one is the dimension, of course, since two vector spaces over the same field are isomorphic if and only if they have the same dimension.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thank you, that's the only one that came to my mind actually !
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:16













2












2








2





$begingroup$

The number one is the dimension, of course, since two vector spaces over the same field are isomorphic if and only if they have the same dimension.






share|cite|improve this answer









$endgroup$



The number one is the dimension, of course, since two vector spaces over the same field are isomorphic if and only if they have the same dimension.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Mar 26 at 14:15









José Carlos SantosJosé Carlos Santos

172k22132239




172k22132239











  • $begingroup$
    Thank you, that's the only one that came to my mind actually !
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:16
















  • $begingroup$
    Thank you, that's the only one that came to my mind actually !
    $endgroup$
    – deeppinkwater
    Mar 26 at 14:16















$begingroup$
Thank you, that's the only one that came to my mind actually !
$endgroup$
– deeppinkwater
Mar 26 at 14:16




$begingroup$
Thank you, that's the only one that came to my mind actually !
$endgroup$
– deeppinkwater
Mar 26 at 14:16

















draft saved

draft discarded
















































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.




draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3163230%2finvariants-between-two-isomorphic-vector-spaces%23new-answer', 'question_page');

);

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







Popular posts from this blog

Adding axes to figuresAdding axes labels to LaTeX figuresLaTeX equivalent of ConTeXt buffersRotate a node but not its content: the case of the ellipse decorationHow to define the default vertical distance between nodes?TikZ scaling graphic and adjust node position and keep font sizeNumerical conditional within tikz keys?adding axes to shapesAlign axes across subfiguresAdding figures with a certain orderLine up nested tikz enviroments or how to get rid of themAdding axes labels to LaTeX figures

Tähtien Talli Jäsenet | Lähteet | NavigointivalikkoSuomen Hippos – Tähtien Talli

Do these cracks on my tires look bad? The Next CEO of Stack OverflowDry rot tire should I replace?Having to replace tiresFishtailed so easily? Bad tires? ABS?Filling the tires with something other than air, to avoid puncture hassles?Used Michelin tires safe to install?Do these tyre cracks necessitate replacement?Rumbling noise: tires or mechanicalIs it possible to fix noisy feathered tires?Are bad winter tires still better than summer tires in winter?Torque converter failure - Related to replacing only 2 tires?Why use snow tires on all 4 wheels on 2-wheel-drive cars?