# Simple way to explain this to an undergrad?

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

1. ### khosaaGuest

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

2. ### Bart GoddardGuest

:

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

3. ### Michael StemperGuest

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
4. ### Michael StemperGuest

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
5. ### Ken PledgerGuest

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
6. ### jbriggs444Guest

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
7. ### Axel VogtGuest

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
8. ### Bart GoddardGuest

(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.

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
9. ### khosaaGuest

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
10. ### khosaaGuest

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
11. ### achilleGuest

achille, Jan 20, 2011
12. ### The Qurqirish DragonGuest

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
13. ### Jeff JohnsonGuest

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