Results are out! Find what you need...fast. Get quick advice or join the chat
x

Unlock these great extras with your FREE membership

  • One-on-one advice about results day and Clearing
  • Free access to our personal statement wizard
  • Customise TSR to suit how you want to use it

Finding out if non-invertible matrices are subspaces of the vector space Mn(R)

Announcements Posted on
Rate your uni — help us build a league table based on real student views 19-08-2015
  1. Offline

    ReputationRep:
    I have Mn(R) is a vector space of all nxn matrices with n>=2.

    Now I need to determine if the set A in Mn(R) is a subspace of Mn(R) where A is the set of all non-invertible matrices.

    Now, I know that in vector fields, to prove that a set is a subspace of a vector field we need to show that the 3 axioms hold:

    (S1) A is not equal to the empty set.

    (S2) A is closed under addition.

    (S3) A is closed under multiplication.

    I also know that a square matrix is non-invertible if its determinant = 0.

    I'm just not really sure how to go about finding out if this set A is a subspace of Mn(R).

    Any hints?
  2. Offline

    ReputationRep:
    As always, the first step is to decide whether or not it's true (then you have to either prove it or find a counterexample).

    If you're not sure, one thing to think about: Why might it be true? If you can't think of any reason for it to be true, maybe it isn't. If you can find a reason for it to be true, that will give you a hint towards how you might prove it.
  3. Offline

    ReputationRep:
    Ok, I've decided that the non-invertible matrices A are NOT a subspace of Mn(R) because it is not closed under addition:

    If you take a non-invertible matrix and add to it another non-invertible matrix, the result is an invertible matrix.

    What's the best way to show this?
  4. Offline

    ReputationRep:
    Would an explicit example suffice? And if so, how do I go about finding two non-invertible matrices?
  5. Offline

    Yes, all you need is one counter-example to show it's not closed under addition.

    To find a non-invertible matrix, think of what other properties it has, as compared to an invertible matrix.
  6. Offline

    ReputationRep:
    Hello again ghostwalker. Properties non-invertible matrices have... their determinants are equal to 0?
  7. Offline

    (Original post by Sun Tower)
    Hello again ghostwalker. Properties non-invertible matrices have... their determinants are equal to 0?
    Yep - should be easy to find a couple and experiment.
  8. Offline

    ReputationRep:
    Ok, so after trying a few it appears that adding two non-invertible matrices DOES give you a non-invertible matrix. Is this always true?
  9. Offline

    (Original post by Sun Tower)
    Ok, so after trying a few it appears that adding two non-invertible matrices DOES give you a non-invertible matrix. Is this always true?
    That's what you have to find out.

    Choose the most basic invertible matrix you can imagine, now try and split it up into the sum of two non-invertible matrices.

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By joining you agree to our Ts and Cs, privacy policy and site rules

  2. Slide to join now Processing…

Updated: March 9, 2009
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

New on TSR

Rate your uni

Help build a new league table

Poll
How do you read?
Study resources
Quick reply
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.