Motivating Matrix Addition/Multiplication Without Appealing to Linear Maps

[Bacle]

Hi, Everyone:

In linear algebra courses, the defs/formulas for
the sum, multiplication of matrices respectively,
are often motivated by the fact that matrix addition
models the point-wise addition of linear maps, i.e.,
If A,B are linear maps described on the same basis, then
the sum (a_ij)+(b_ij) describes the linear operator:

(A+B)(x)=A(x)+B(x)

And AB models the composition of the operators A,B;
i.e., A*B(x) =A( B(x)).

Now, I am teaching a class in which matrices have,
so far, been used only to represent systems of linear
equations. Does anyone know how to motivate the
definitions A+B and AB from this or a related
perspective?

Thanks.

 

[AlephZero]

You could use the same basic ideas expressed as equations.

If Ax = p and Bx = q, then (A+B)x = (p + q)
If Ax = y and By = z, then BAx = z

The addition example seems a bit artificial, but most students will probably "buy" matrix addition by analogy with scalar addition.

The multiplication example is (fortunately) more realistic. You can invent "word problems" where two sets of equations can be combined and solved this way.

 

[Simon_Tyler]

The addition is not that artificial - you rewrite some of the equations in the linear system - then add/subtract from your original system to get a simplification.

 

[sponsoredwalk]

I'll write the number 2 as 02, and 3 as 03 etc.. just to make things look pretty on this page.

I think you could begin by explaining how the matrix on the left comes from considering that
system of equations on the right. You could motivate the definition by explaining how a
matrix is an abstract representation of that system of equations first.

|01 02 03 04| - 01x + 02y + 03z + 04w
|05 06 07 08| - 05x + 06y + 07z + 08w
|09 10 11 12| - 09x + 10y + 11z + 12w
|13 14 15 16| - 13x + 14y + 15z + 16w

You get the idea, the - on every line is typographical...Then explain that an equation like 01x + 02y + 03z + 04w could be perceived as coming
from adding two different equations as follows:

_00x + 01y + 01z + 02w
+01x + 01y + 02z + 02w
------------------------
(00 + 01)x + (01 + 01)y + (01 + 02)z + (02 + 02)w

so

(00 + 01)x + (01 + 01)y + (01 + 02)z + (02 + 02)w = 01x + 02y + 03z + 04w.

We can rewrite the whole system in this way:

|(00 + 01) (01 + 01) (01 + 02) (02 + 02)| - (00 + 01)x + (01 + 01)y + (01 + 02)z + (02 + 02)w
|(03 + 02) (03 + 02) (03 + 04) (04 + 04)| - (03 + 02)x + (03 + 03)y + (03 + 04)z + (04 + 04)w

I won't do all four as you get the idea.

So we have

|01 02 03 04| = |(00 + 01) (01 + 01) (01 + 02) (02 + 02)| = |00 01 01 02| + |01 01 02 02|
|05 06 07 08| = |(03 + 02) (03 + 02) (03 + 04) (04 + 04)| = |03 03 03 04| + |02 03 04 04|

and you can see that the definition of matrix addition follows. Obviously the last equality
should be approached starting from the perspective of two systems of equations where
you just show they have the same solution and show that there's no reason not to define
things this way because everything has the same solution set...

As for matrix multiplication that's a big question, you might enjoy reading my thread here:
https://www.physicsforums.com/showthread.php?t=451822
as I tried to figure this question out and eventually got a few separate and equally
satisfying answers. From post 6 on is where I wrote the ideas, the early posts are just
me being confused.