(1) NO CLASS FRIDAY 10/17 and special assignment. There will be no class on Friday, 10/18. In place of class is the special assignment to view online the video of Lecture 6 (Column Space and Nullspace) of Gilbert Strang's Linear Algebra course at MIT. There is a link to the webpage for these videos linked to our class website. You can also find the lectures free in the ITunes Store.
(2) Midterm #1 FRIDAY 10/24. This exam will cover what we have done in Chapter 1, plus Sections 3.1, 3.2, 3.3. More detail will be spelled out next week but you should keep this in mind as you study this week.
This is a very short section that is a reminder of the geometry of lines and planes in two-space and three-space as they are described by algebra. This should be familiar ground.
One key thing to notice in this section is the 'set notation' that is used in later sections. For example, in problems 12-17.
(1) On pages 167-8 are given the algebraic properties of vector addition and multiplication of a vector by a scalar (a real number). This is not new since it is a special case of the algebra of matrices, so this should not be a major focus.
(2) At the bottom of page 168 begins the subsection on Subspaces of Rn. This is a BIG DEAL, but the book is kind of sneaky about the definition. The official definition is a subset S of Rn that satisfies all the 10 properties of Theorem 1, but the practical definition is that this is automatic if you check the 3 properties in Theorem 2. We will use Theorem 2 as our definition. You will use Theorem 2 in problems to check that sets are subspaces.
(3) Study Example 1. It shows how the set of solutions to a homogeneous linear equation are a subspace. See how addition and multiplication is checked for general solutions u and v, not for numerical examples.
(4) On page 171 is a step-by-step flow chart or list of instructions on how to check a set is a subspace. Study how this is used in Examples 2 through 5. Look for patterns. They will be spelled out in Section 3.3.
(5) Study some of Exercises 9-25 (odd) and the answers in the back of the book.
The key to this section is noticing that there are basically two ways that a set of vectors in this section is specified as either: (a) the vectors are given by a formula in parameters that are free variables or (b) the set is the set of solutions to some linear equations.
(1) Definition: The span Sp{F} of a subset of vectors F is the set of all linear combinations of the vectors in the set. On page 176 this is shown to be a subspace always, for any F. Then the trick is to recognize a span when you see one. For example, if A1, A2, A3 are the columns of a matrix A, then the set of vectors b for which Ax = b is consistent is the span of the columns of A. This is because Ax = b can be rewritten as x1A1 + x2 A2 + x3A3 = b.
(2) The span of the columns of a matrix A is called either the Column Space of A or the Range of A (see pp.181 – 183). In the text it is denoted as R(A). Since the Range is defined to be a span, it is a subspace by Theorem 3. Notice that if A is an mxn matrix, the Range is a subspace of Rm.
(3) Definition: The Null Space of a Matrix A is the set of x for which Ax = 0. It is proved to be a subspace in Theorem 4. In the text it is denoted as N(A). Notice that if A is an mxn matrix, the Null Space is a subspace of Rn.
(4) The Row Space of a matrix A can be viewed as the Range of the Transpose of A, with the vectors written as row vectors and not column vectors. Notice that if A is an mxn matrix, the Row Space is a subspace of Rn.
(5) All of these spaces are subspaces. What are not subspaces that arise in linear equations are these main examples:
a. NOT A SUBSPACE: The solution set of Ax = b, where b is not the zero vector. In other words, the solutions to inhomogeneous equations. If this equation is consistent, this is not a subspace since the 0 vector is not in the set, and the sum of two solutions is a solution of Ax = 2b, so the sums of vectors are not in the set either. This set of solutions is a translation of the subspace that is the solution set of Ax = 0.
b. NOT A SUBSPACE: A set of vectors of the form A0 + c1A1 + c2 A2 + . . . + cnAn, where scalars ci vary freely but the vector A0 is not the zero vector. Note that again the 0 vector is not in the set and that sums are not in the set.
(6) Study the Exercises 1 – 19 (odd) and try to spot whether or not you can write the ones that are subspaces at either a Range or a Null Space of some matrix. Or if the set is not a subspace, check to see whether it is one of the cases of (5) above.
(Show your work; do the problems by hand except as noted.)