MWF 4:40-5:30 pm, 167 Willard

office hours: MWF 9:05-11:00 am. 205
MB

outlines & grading

integrity |

textbook: *Linear Algebra with Applications *by Bretscher,
Prentice Hall 2001.

100% = 272 pts on Dec. 13.

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**First class (August 28):**

Examples of linear systems.

What is a solution?

What is a linear equation?

What are equivalent systems? (the same solutions)

Solving ax=b for x.

**Second class (August 30).**

Why they are called linear?

Solving an arbitrary system of linear equations.

**Sept 4.**

Answer in standard form (no changes of variables). Three cases:

0=1 (no solutions).

the variables = numbers (unique solution).

distinct variables = linear functions in the other variables (at least
one) + numbers (infinetely many solutions).

We do not use echelon forms. We drop zero columns and zero rows.

Examples.

Matrix addition and multiplication. Matrix representation of a system
of linear equations, *Ax=b*.

Homework, due Friday, 4:40 pm. Exercises 1--5 in Section 1.1; 5 problems,
5 pts each.

Linear combinations and relations.

Linear dependence and independence.

**Sept 6. **Chapter 2.** **Linear and affine transformations.

Linear and affine transformations of plane.

**Sept 9.** Invertible affine transformation.

The transformation x -> Ax+b is invertible iff the matrix
A is invertible.

The inverse, if exists, is an affine transformation.

To prove the "only if" part we simplify the transformation composing
it with invertible

transformations. First we drop b, then we make A an identity
matrix possibly augmented by zero columns and/or columns. This part does
not use the assumption that transformations is invertible.
Using the assumption,

we see that there are no zero rows or columns, i.e., A is the

identity matrix. Now it is clear that A is invertible.

Homework was given, 25 pts due W.

**Sept 13. ** Kernel and image of a linear transformation.
Dimension of vector subspace.

**Sept 16.** Bases, coordinates. Vector spaces.

Homework was given, 20 pts due W. Ex. 2 from each of 4 sections in
Chapter 3.

**Sept 18.** Chapter 4.

The row operations do not change the row space and relations between
the columns.

The column operations do not change the column space and relations
between the rows.

So the row and column do not chage the dimensions of the column
and row space of

a matrix. Therefore, both dimesions are the same (and equal to the
rank

of the matrix).

**Sept 25.** W. midterm .
min = 10, average = 32, max = 50.

**Sept . 27.** F. Midterm discussion. The grader thought that it
was 5 pts per problem, so scores

for the midterm should be doubled. Many students still cannot solve
ax=b (see the first class).

rank(diagonal matrix) = # nonzero entries.

**Sept.30**. M. Chapter 5. Homework due W, 5 problems, 5 pts
each. page 190. #2, 4, 6, 8. Page 199., #10.

h4: min=15, av.=23, max=25 pts.

**Oct 7.** Homework due W, Oct 9: page 222: 16,18,20,22,26.

5 problems, 5 points each.

October 30. W midterm 2. ps | pdf | pictures. max=46. min=16. average=29.

M, November 25, midterm 3. ps | pdf | pictures

M Dec 2, hmw 10 pts due, #35 on page 374. Hints:

compute the eigenvalues of the *m* by *m* matrix
with all entries 1 (most of them are zeros);

(answer: since the rank of this matrix is 1, only one eigenvalue
can be nonzero; since the trace is *m*, this eigenvalue is *m*);

consider the Gram matrix (the matrix of all dot products of given *n*+1
vectors).

Since the vectors are linearly dependend, the determinant of the Gram
matrix is 0.

W, Dec. 4. We did Section 9.2.

The derivative (*e ^{A}*)' of

when