This is a key section, but it will go well if you notice that the "rules" have changed a bit! Up to now, the work has centered on ways of carrying out procedures, such as solving linear equations and multiplying matrices. In this section the key points are two concepts spelled out by mathematical definitions: linear independence (or dependence) of vectors and singularity (or non-singularity) of square matrices.
Math definitions are not just descriptions; they are prescriptions. If you are asked whether a set of vectors is linearly independent, then the answer comes from checking whether the definition is satisfied. As the section goes on, we learn some methods for making this check, but the answers to everything go back to the definitions. These definitions are not optional and not side items, they are the center of what is going on. So you need to understand every word and then be able to reproduce and use every word.
You will definitely need some more practice with these concepts than just the problems assigned to turn in. But the good news is that in the end solving the problems and answering the questions comes down to reducing matrices and answering questions about solutions of linear equations. But the questions are different even if the methods are the same.
For practice, you should do some additional odd problems from 1-15, 17-27 and 29-33 until they become routine exercises that you are confident that you can do 100% of the time (check the answers in the back of the book). For more conceptual challenges, try 51 and 55.
We will only look at the data fitting pages of this section, not the numerical integration and differentiation. You have already done one problem fitting a quadratic function to data points, with graph a parabola. This is a general discussion of the same idea, with more points and higher degree polynomials, as well as some problems with exponentials.
The interpolation problems reduce to a problem of solving some linear equations, and the question of always being able to solve, and not just doing individual examples, depends on establishing the non-singularity of the Vandermonde matrix. Read carefully and see how this is established by counting the roots of a certain polynomial.
For practice with polynomial data fitting, you can try 1, 3 or 5. But it would be a good idea to try 7 or 9 to be sure you understand the exponential data fitting also.
This section answers the question: If we can multiply matrices, can we also divide? The answer is Òsometimes, sort ofÓ but the details will fill in a key missing piece of the Ax=b story by introducing A-1, which exists for some matrices. So we will learn the definition and when the inverse exists, the algebraic properties and also how to compute the inverse.
You will really need to practice finding the inverse in some of the Exercises 13-21. Can you explain the singularity of matrices in problems 9 and 11 without going through a long algorithm? The problems 35-45 are short but very good practice to get solid on the algebraic properties.
Familiarize yourself with the formula for inverses of 2x2 matrices. If not memorized, you can learn it close enough to re-derive it as needed.
(Show your work; do the problems by hand except as noted.)