Simple way to explain this to an undergrad?

Discussion in 'Undergraduate Math' started by khosaa, Jan 19, 2011.

  1. khosaa

    khosaa Guest

    Hi,

    I am wondering if anyone can give a simple way to answer the following
    question in a way that is understandable to a first or second year
    college student (i.e. one who has not taken a course in abstract
    algebra).

    Typically students are taught two the following 2 basic facts about
    matrices:
    Fact 1 : For A, a matrix of size mxp and B, a matrix of size rxn, the
    matrix product AB is defined iff p=r.
    Fact 2 : scalars can be considered as 1x1 matrices
    Given these two facts, the question arises as to how we can say that
    the products cA and Ac are defined (if m ne 1 and p ne 1 resp for the
    matrix A described above).

    One way around this is to simply define scalar multiplication (as it
    is normally defined), but this still leads this nagging doubt as to
    how one resolves this apparent inconsistency that exists with the
    aforementioned 2 facts.

    Sorry for this seemingly stupid question, but it just bugs me a bit.

    Thanks,
    Fran Khossa
     
    khosaa, Jan 19, 2011
    #1
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  2. khosaa

    Bart Goddard Guest

    :

    My gut reaction is to just say "hey, it's two different
    operations." But another tack, which might satisfy the
    student until he knows better is to say that "cA"
    (scalar mult) is really just shorthand for "(cI)A"
    (matrix multiplication.)
     
    Bart Goddard, Jan 19, 2011
    #2
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  3. This is where you go off the tracks. The reals are isomorphic to the
    1x1 real matrices, but they are not the same thing. Multiplication of
    a scalar times a matrix is a different operation than multiplication
    of two matrices.
     
    Michael Stemper, Jan 19, 2011
    #3
  4. I think that you should go ith your gut.
    That doesn't help, because it just changes the question from "how do
    you multiply cA to how do you multiply cI?"
     
    Michael Stemper, Jan 19, 2011
    #4
  5. khosaa

    Ken Pledger Guest

    Don't say that. You could mention that Fact 1 and the definition of
    addition make 1 x 1 matrices combine *with one another* as scalars do
    (to put it simply), but there's no need to press it further.

    The only place where I find "Fact 2" notationally useful is with
    bilinear or quadratic forms. If X = (x_1 x_2 x_3) and A is 3 x 3,
    then it's convenient to write the equation XA(X^T) = (0) in the more
    slipshod form XA(X^T) = 0. By the time students reach that stage,
    the abbreviation doesn't seem to confuse them.

    Ken Pledger.
     
    Ken Pledger, Jan 19, 2011
    #5
  6. khosaa

    jbriggs444 Guest

    I don't know about that. The operation of going from
    scalar c to square matrix cI is not neccessarily done
    by scalar multiplication. I took it as an instance
    of economical notation without an implied implementation.

    It could just as easily be implemented by populating
    an n by n square matrix with c along its main diagonal.

    Whether the idea of treating a scalar as denoting
    its appropriately sized square matrix equivalent is
    actually a useful way of thinking about scalar
    multiplication -- I don't know.
     
    jbriggs444, Jan 19, 2011
    #6
  7. khosaa

    Axel Vogt Guest

    I am 'old' and not used what 'undergrad' actually means ...
    we learned 2*2 and 3*3 matrices at school (what a pain) and
    at the Uni in the 1st year.

    And never understand, why one would say such to such to the
    students in their first course.

    Either you do it 'operationally' (well some are just forced
    to pass some Math) and say: 'try to apply the rule' (what a
    terrible drill).

    Or a bit further with care: lin Alg = solving equations and
    how to understand 'mappings'.

    Compositions of maps and a simple picture/notation will never
    need 'fact1'. And 'fact2' only says, that the lecturer is not
    able to explain, what a linear space is or what a linear equs
    are.

    What a sad curriculum: drilling students without ever giving
    them a chance to understand.
     
    Axel Vogt, Jan 19, 2011
    #7
  8. khosaa

    Bart Goddard Guest

    (Michael Stemper) wrote in
    No. The original question wasn't "how to do you multiply cA",
    but "how do you explain to a student that writing cA doesn't
    violate the matrix multiplication rule about sizes." Often,
    students ask questions that are too hard for them. Sometimes
    it's best to just give them something "sort of true" to tide
    them over till they know better. Further, we don't have to
    multiply cI. This is just me using a text-based notation for
    the matrix with c's on the main diagonal and zeros elsewhere.
    There's a matrix action which scales all the row vectors by c.
    The matrix which does this is cI. We abbreviate this action
    by writing cA.

    This is similar to answering a toddler's questions about
    where babies come from. You can tell them a tiny true
    thing (like "mommies' tummies") and they're completely
    satisfied for many years. A long dissertation about
    the functions of genitalia would do no good.
     
    Bart Goddard, Jan 19, 2011
    #8
  9. khosaa

    khosaa Guest

    I think there are times when it is useful. I can't recall the exact
    application, but I know it comes in handy in performing some necessary
    matrix manipulations necessary in statistics.
     
    khosaa, Jan 20, 2011
    #9
  10. khosaa

    khosaa Guest

    Thank-you for the clarification. So the best thing to do is not even
    state "fact 2", sice it is not a fact. Considering the fact that it
    adds little to the students understanding that is useful, it is best
    to simply stay clear of the whole issue.

    Fran
     
    khosaa, Jan 20, 2011
    #10
  11. khosaa

    achille Guest

     
    achille, Jan 20, 2011
    #11
  12. As others have stated, "Fact 2" shouldn't be used. I never learned to
    associate scalars with 1x1 matrices.
    If you insist on relating back to older concepts, rather than just
    defining scalar multiplication of matrices, then I suggest you use the
    distributive rule.
    You can analogize this was:
    multiplication is distributive over addition: a (b + c) = ab + ac. You
    take each term in the sum and multiply it by the constant value
    outside.
    Similarly, for cA, you multiply everything inside the matrix by the
    constant value outside. Multiplication "distributes" over a matrix's
    elements
     
    The Qurqirish Dragon, Jan 20, 2011
    #12
  13. khosaa

    Jeff Johnson Guest

    It's called context and there is no doubt. They are two different operations
    that are both well defined and lead to no inconsistencies. Is there any
    doubt that a*V is valid when a is a scalar and V is a vector? If you don't
    question that then you shouldn't question it if V was a matrix.

    Scalar multiplication is a short hand for multiplying all the elements(which
    generally are scalars) by the scalar.

    a*B is "shorthand" for

    [a*B11 a*B12 a*B13 .... ]
    [ ... ]
    [a*Bn1 a*Bn2 a*Bn3 .... ]

    A*B is not "shorthand" for

    [A*B11 A*B12 A*B13 .... ]
    [ ... ]
    [A*Bn1 A*Bn2 A*Bn3 .... ]

    Although if you wanted you could define it that way... no one would get it
    unless you told them that is what you mean and then they would still
    probably get confused if you used the same symbol.

    Again though, why make it complex. You could have asked the same question
    about vectors. If the "student" doesn't get confused about scalar
    multiplication in a vector space then he shouldn't get confused with
    matrices. If he's getting confused about it with matrices then he doesn't
    really understand it vectors.
     
    Jeff Johnson, Jan 21, 2011
    #13
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