At last some completely new ideas! This is all about right angles and sets of vectors that are orthogonal (i.e. perpendicular) coordinate vectors. The key idea is that two vectors are orthogonal if their dot product equals zero. Then we use a lot of linear algebra ideas.
Some key facts: (1) An orthogonal set of non-zero vectors is always linearly independent. (2) There is a formula involving dot product that will solve the equation Ax = b, if the columns of A are orthogonal. (3) There is a method called Gram-Schmidt to convert any linearly independent set into an orthogonal set with the same span. Examples of these points were presented in the Wed Nov 5 class.
DO odd problems 1-7.
In this section is defined a special kind of function T from Rn to Rm called a linear transformation. For such a function it is possible to define a null space and a range.
Then we discover that any linear transformation T is really a matrix functions with this kind of formula: T(X) = AX for some matrix A. And the null space of T is the null space of A and the range of T is the range of A.
So what is the point of adding this definition? If we change coordinates, the function stays the same but the coordinate formula changes, for the formula now has a different matrix B. Figuring out the relationships among the various formulas for T is based on the properties of linear transformations.
The ÒDOÓ section is really
important this time, since these problems are all really easy but really
important and there is no reason to turn them in, but there is a reason to
practice with a variety of examples.
DO: A large sampling of the odd
problems 1 – 17.
(Show your work)