Welcome to Mathematics Stack Exchange

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's built and run by you as part of the Stack Exchange network of Q&A sites. With your help, we're working together to build a library of detailed answers to every question about math.

We're a little bit different from other sites. Here's how:


Ask questions, get answers, no distractions

This site is all about getting answers. It's not a discussion forum. There's no chit-chat.

Just questions...

...and answers.

up vote

Good answers are voted up and rise to the top.

The best answers show up first so that they are always easy to find.

accept

The person who asked can mark one answer as "accepted".

Accepting doesn't mean it's the best answer, it just means that it worked for the person who asked.

linear map $f:V \rightarrow V$, which is injective but not surjective

up vote 14 down vote favorite

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective.

I always thought that if the dimension of the domain and codomain are equal and the map is injective it implies that a map is surjective. Maybe we need an infinite basis of the vector space $V$? What can be an example of that?

Thank you!

2 Answers

up vote 4 down vote accept

Yes, we need an infinite-dimensional vector space. An interesting example is: $V$ the space of continuous functions $[0,1]\to\mathbb R$ and $f$ integration $f(g)(x)=\int_0^xg(t)\,\mathrm dt$. This is not surjective because $f(g)(0)=0$ for all $g$

up vote 3 down vote

In finite dimensions we have that bijectivity $\Leftrightarrow$ injectivity $\Leftrightarrow$ surjectivity. Hence we have to come up with an infinite-dimensional example. The idea is to pick a basis $v_i, i\in \mathbb N$, and shift every basis vector $v_i \mapsto v_{i+1}$. We can do that not hitting the first basis vector only because we have infinitely many elements.


Get answers to practical, detailed questions

Focus on questions about an actual problem you have faced. Include details about what you have tried and exactly what you are trying to do.

Ask about...

  • Understanding mathematical concepts and theorems
  • Hints on mathematical problems (but please read our FAQ about homework questions)
  • History and development of mathematics
  • Solving mathematical puzzles
  • Software that mathematicians use

Not all questions work well in our format. Avoid questions that are primarily opinion-based, or that are likely to generate discussion rather than answers.

Questions that need improvement may be closed until someone fixes them.

Don't ask about...

  • Anything not directly related to math
  • Questions that are primarily opinion-based
  • Questions with too many possible answers or that would require an extremely long answer
  • Physics, engineering and financial questions.
  • Questions about typesetting equations
  • Numerology questions
  • Questions about the site itself — please ask on meta instead

Tags make it easy to find interesting questions

All questions are tagged with their subject areas. Each can have up to 5 tags, since a question might be related to several subjects.

Click any tag to see a list of questions with that tag, or go to the tag list to browse for topics that interest you.

linear map $f:V \rightarrow V$, which is injective but not surjective

up vote 14 down vote

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective.

I always thought that if the dimension of the domain and codomain are equal and the map is injective it implies that a map is surjective. Maybe we need an infinite basis of the vector space $V$? What can be an example of that?

Thank you!


You earn reputation when people vote on your posts

Your reputation score goes up when others vote up your questions, answers and edits.

+5 question voted up
+10 answer voted up
+15 answer is accepted
+2 edit approved

As you earn reputation, you'll unlock new privileges like the ability to vote, comment, and even edit other people's posts.

Reputation Privilege
15 Vote up
50 Leave comments
125 Vote down (costs 1 rep on answers)

At the highest levels, you'll have access to special moderation tools. You'll be able to work alongside our community moderators to keep the site focused and helpful.

2000 Edit other people's posts
3000 Vote to close, reopen, or migrate questions
10000 Access to moderation tools
see all privileges

Improve posts by editing or commenting

Our goal is to have the best answers to every question, so if you see questions or answers that can be improved, you can edit them.

Use edits to fix mistakes, improve formatting, or clarify the meaning of a post.

Use comments to ask for more information or clarify a question or answer.

You can always comment on your own questions and answers. Once you earn 50 reputation, you can comment on anybody's post.

Remember: we're all here to learn, so be friendly and helpful!

up vote 9 down vote

Yes, we need an infinite-dimensional vector space. An interesting example is: $V$ the space of continuous functions $[0,1]\to\mathbb R$ and $f$ integration $f(g)(x)=\int_0^xg(t)\,\mathrm dt$. This is not surjective because $f(g)(0)=0$ for all $g$

edit

@marco11: $f$ is injective, because if $g_1$ and $g_2$ differ at some point -- say $g_1(x_0)\ne g_2(x_0)$, then $f(g_1)$ and $f(g_2)$ will have different derivatives at that point, and so cannot be the same function. On the other hand $f$ is not surjective because, for example, the function $x\mapsto x+1$ is not $f(g)$ for any $g$ (namely $f(g)(0)=\int_0^0 g(t)dt = 0 \ne 0+1 $). - Henning Makholm May 24 at 22:42

add comment


Unlock badges for special achievements

Badges are special achievements you earn for participating on the site. They come in three levels: bronze, silver, and gold.

In fact, you can earn a badge just for reading this page:

 Informed Read the entire tour page
 Student First question with score of 1 or more
 Editor First edit
 Good Answer Answer score of 25 or more
 Civic Duty Vote 300 or more times
 Famous Question Question with 10,000 views

see all badges


Find a question to answer, or ask your own

Looking for more in-depth information on the site? Visit the Help Center

Mathematics Stack Exchange is part of the Stack Exchange network

Like this site? Stack Exchange is a network of 143 Q&A sites just like it. Check out the full list of sites.

Stack Exchange