Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

smr_kml Group Title

How to know if the matrix is on-to and one-to-one?! Is there an easy way to know that?!

  • 2 years ago
  • 2 years ago

  • This Question is Closed
  1. cibychak Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    onto and one to one are not props of a matrix.they belong to functions

    • 2 years ago
  2. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    talking about matrix transformations I assume

    • 2 years ago
  3. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    it also belongs to matrix in linear algebra!

    • 2 years ago
  4. Jemurray3 Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    That's not true, you can assign such properties to any transformation.

    • 2 years ago
  5. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    http://www.mast.queensu.ca/~spencer/apsc174/11onto.pdf

    • 2 years ago
  6. cibychak Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    oh.thanks guys,i'm just a high school level mathematician :)

    • 2 years ago
  7. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    @TuringTest Can you explain it to me?!

    • 2 years ago
  8. Jemurray3 Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    @TuringTest it's all yours :)

    • 2 years ago
  9. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    I would just go down the checklist for a given transformation is n<m ? if so then not onto if the column space is \(\mathbb R^m\) it is onto etc. I was really hoping you'd take it over @Jemurray3; I've never even heard of the term "onto" until recently

    • 2 years ago
  10. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    I also have class in about 5 min...

    • 2 years ago
  11. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Sorry for wasting your time! @TuringTest

    • 2 years ago
  12. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    not wasted at all, I'll be happy to see other's explanations and save the medal for someone who deserves it !

    • 2 years ago
  13. slaaibak Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    to check one-to-one, just check if it has only the trivial solution. that is x=0 is the only solution to Ax=0

    • 2 years ago
  14. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    So when it has only solution Ax = 0, it will be one to one?!

    • 2 years ago
  15. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    sorry no

    • 2 years ago
  16. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    ok and if n<m so it's not onto? right?

    • 2 years ago
  17. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    the null space must be zero, so no solutions

    • 2 years ago
  18. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    I get confused

    • 2 years ago
  19. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    that part is right Ax=0 has not solutions but the trivial one if A is invertable if A is invertable the transformation is one-to-one

    • 2 years ago
  20. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    has no*

    • 2 years ago
  21. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    invertible means the det. is not equal zero?

    • 2 years ago
  22. slaaibak Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    the dimension of the nullspace should be zero...I think even if the dimension of the nullspace is zero, it still contains the zero vector and hence the nullspace can't be zero.. or am I wrong?

    • 2 years ago
  23. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    right it also implies like 12 other equivalent facts about the matrix

    • 2 years ago
  24. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    Nullspace is the dimension of the solution space for Ax=0 and the zero vec has no dimension, so I think that is wrong slaaibak I make a lot of mistakes with this stuff too though

    • 2 years ago
  25. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    :(

    • 2 years ago
  26. slaaibak Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Nullspace is the solution space to Ax=0. nullity is the dimension of the nullspace

    • 2 years ago
  27. Jemurray3 Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Then I shall supplement: If we have an nxm matrix, it maps m-dimensional vectors into n-dimensional space. If our transformation is onto, that means that every vector in n-dimensional space is the image of some vector in m-dimensional space under our transformation. Put another way, if we call our matrix A, then we have at least one m-dimensional vector x such that Ax = b for any vector b in R^n. To check for this, try to see if the (m) columns of A span n-dimensional space. I assume you are familiar with the concept of spanning sets, so I'll just say that obviously we must have that m >= n (a necessary, but not sufficient condition). We also require that at least n of the m columns are linearly independent. A one-to-one transformation as stated above means that every vector in m-dimensional space is associated to one and only one vector in n-dimensional space. Therefore, we should be able to go from one vector to another and back with no ambiguity, i.e. the transformation is invertible. For this, we'd need a square matrix whose determinant is not equal to zero.

    • 2 years ago
  28. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    right

    • 2 years ago
  29. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    ah damn

    • 2 years ago
  30. AravindG Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    am i late?

    • 2 years ago
  31. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    No

    • 2 years ago
  32. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    Good explanation JM I'm off to class somebody take away my medals and give them to slai and JM, I don't know why I have the most :/ later guys!

    • 2 years ago
  33. AravindG Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    thats nic of u @TuringTest

    • 2 years ago
  34. slaaibak Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    correct me if I'm wrong, but a 3 * 2 matrix transformation can also be one-to-one? so checking if its invertible is not the only criteria

    • 2 years ago
  35. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    if it's invertible so it's one-to-one?

    • 2 years ago
  36. slaaibak Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    non-square matrices aren't invertible.

    • 2 years ago
  37. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    I know that and if the det is = 0 so it's not invertible

    • 2 years ago
  38. Jemurray3 Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Invertible matrices are one-to-one, yes. And @slaaibak can you come up with an example of one?

    • 2 years ago
  39. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    how about onto?

    • 2 years ago
  40. slaaibak Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    1 2 2 1 2 1

    • 2 years ago
  41. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    This is not one - to -one right?!

    • 2 years ago
  42. slaaibak Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Why not? Solve it for Ax=0

    • 2 years ago
  43. Jemurray3 Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    It just so happens that invertible matrices are also onto, but a matrix doesn't necessarily HAVE to be invertible in order to be onto.

    • 2 years ago
  44. smr_kml Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    how can I know if it's onto?

    • 2 years ago
  45. slaaibak Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Check if it's consistent for Ax=b. for every b?

    • 2 years ago
  46. Jemurray3 Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    @slaaibak good example. Invertibility is a sufficient, but not necessary condition. I'll revise to say linear independence of columns.

    • 2 years ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.