This is a section devoted to applications of the linear algebra in sections 1.1-1.3. There are two related kinds of problems, network problems and electrical circuit problems. In the class we will focus on the network problems, but if you are in a field that uses Ohm's Law, you may want to spend some extra time understanding those examples as well.
For the network problems, if you are interested in applications, or to be sure you understand the method, you can study the examples and practice with problems 1 and 3. For the circuit problems – if you are interested – you can do the same.
This is a key section with lots of new definitions and concepts. The central ideas are the new concepts of addition and multiplication with matrices, giving the basics of matrix algebra. But also, there is a really important wrinkle added to the linear equation solution story; this is the vector form for the solutions (same solutions but a new way of looking at them). This is very, very important for us in the course.
This section calls for a lot of practice at various levels. First of all, you want to be completely confident that you can perform accurately all the matrix operations. To do this, you really must practice some examples, such as the odd problems from 1-11 but even more the odd problems 31-41. For the latter problems, slow down and ask yourself what the problem is illustrating (for instance, what is happening in 31, what are the "shapes" of the matrices in 33 and 41). By shape I mean the number of rows and columns.
Also, be sure to practice with the odd problems 43 – 49 to write down solutions in vector form (these problems require little computation, happily). Some of the non-computational problems are also instructive. For instance, 53 requires no computation, just thinking and understanding matrix multiplication. Likewise 55.
This section is much shorter than 1.5, but it also has some key aspects of matrix algebra, namely what are the rules? And in what ways is matrix algebra like the algebra with numbers and in what ways different. Also, the concept of the transpose is introduced, along with symmetric matrices. Finally the dot or scalar product is defined and used to compute the length of a vector.
The basic definitions and properties appear in the computations in exercises 1 – 25, and you should do a selection at least (ask yourself what property or concept showing up in the problem). You will also learn a lot wrestling with a couple of the more conceptual problems 27 – 33.
(Show your work)
*Special Instructions
for 1.5 #57
For 1.5 #57, you are supposed to compute P2X.
If the arithmetic gets too ugly for you, do this: write down the expressions to be computed as sums of products (or at least some of them) without evaluating them. For example, you can write 1*2 + 2*7 + 5*3 but without computing the sum 31. Then you can do the arithmetic with the calculator. But the key thing is to make very clear what computations you are performing.
*Special Instructions
for Section 1.6, #26.
If you find a counterexample, don't stop there. Give a brief explanation of what goes wrong to make the statement untrue (and thus show for what cases it might go right).