Section 1.4 key concepts:

	Matrix-vector multiplication, definition on page 35.

	If A is a matrix and x is a vector, make sure that you understand
	the concept of SPAN and linear combination from the prior section
	so well that it becomes obvious to you that:

		A x

	is a linear combination of the columns of A.  What if this is
	not obvious?  The way to learn the meaning of each concept is
	by writing down examples:  Try to do the following

		Write down a very simple example a1, a2, b  with b in SPAN(a1,a2).

	Then:
		Write down a very simple example where b is not in SPAN(a1,a2).


	Similarly: Write a very short example for Theorem 3 (just pick
	some small a1,a2,b  and try to understand what Theorem 3 is saying
	just for that particular example).

	Likewise, for theorem 4, write down a very small/simple matrix
	for which statements a,b,c,d are obviously true, and also a
	simple matrix for which a,b,c,d are obviously false
	(e.g. a 2x2 matrix in which the second row is zero.  Or even
	simpler, a 1x1 matrix that is identically zero).  Then try
	to understand how a,b,c,d relate to each other in your examples.

	Similarly, to understand Theorem 5, you could pick some
	very small examples, e.g. A could be a 1x2 matrix (i.e. just 1 row),
	for such examples it is easier to see why Theorem 5 is true.


Exercises in 1.4:

I assume that Ex 1-10 are easy?  (if so, we can skip them, if not,
please let me know on Thursday!)

Exercises 11-12 are just like Exercises 13-14,  so while you should
definitely be able to do Ex 11,12,   it seems that it would be OK
to just to Ex 13 and 14.

Lets turn in one of Ex 13 or 14.
And turn in one of Ex 15 or 16.
So that's two turn-in exercises in total.

Next, read Ex 17-25.  You don't need to turn these in, but if you see any
that you're not sure that you can solve, please ask on Thursday!