# Linear Algebra

**Due Today.** Exercises from section 1.1, page 8,

numbers 1–4, 13–14, 21–22, 23, and T4.

**Read for Friday.** Finish 1.2 on matrices and

read 1.3 on dot products and matrix multiplication.

**Due Friday. **Exercises from section 1.2: 1–2, 4–7

parts a–d each, 8, 9, T1, T5a.

**No class Monday. **Martin Luther King Jr. Day.

**Due Wednesday.** Exercises from section 1.3: 1–

4, 7, 9, 11–12, 19–20, 33, T1, T4, ML1, ML2, ML5,

ML6ac.

The ML exercises are Matlab exercises. Do

them, but you don’t have to make printed versions

of them. Feel free to use Matlab to help you

with your homework, even the parts of Matlab

we haven’t discussed in class.

**Last time.** We discussed systems of linear equations.

Sometimes they’ve got exactly one solution,

sometimes no solutions, and sometimes infinitely

many solutions. We saw the method of elimination

as the ancient Chinese did it, how they created a

matrix from the system of linear equations , first

put the matrix into echelon form, then into reduced

echelon form, and then they could read the solution

right from the resulting matrix.

We’ll formalize this method soon.

**Today. **We’ll review some of the topics in section

1.1 about linear systems , and we’ll begin discussing

section 1.2 which introduces matrices.

**Standard notation and terminology.** When

we’ve got a system of linear equations, we’ll usually

let m denote the number of equations , and n

the number of variables, and say the system is a

m× n system of equations. For instance, here’s an

example 3 × 2 system of equations:

Here, m = 3 since there are 3 equations, and n = 2

since the two variables are x and y . (Typically,

when a system has more equations than unknowns ,

then there are no solutions. Does this system have

any solutions?)

When there are many variables, we usually subscript

them, but when there are few variables we

ususally use different letters for the variables. For

instance, if n = 6, we might use the 6 variables

, and ,
but if n = 3, we might use

the three variables x, y, and z.

A general m × n system looks like this :

where the n variables are
. The m

constants that appear on the right side of the m

equations are denoted . Since each of

the m equations has n coefficients for the n variables,

there are mn coefficients in all, and these are

denoted with doubly subscripts, so, for instance

is the coefficient of the variable in the
third equation.

A solution to the system is an n-tuple of values

for the n variables that make all the equations simultaneously

true. For example, in the 3×2 system

above, a solution is (x, y) = (2, 1). In fact, it’s the

only solution for that system.

**Matrices.** A matrix is a rectangular array of

numbers. You can see how they’re related to systems

of equations since the system is determined

by its coefficients and its constants, and those can

naturally be displayed in a matrix. Consider the

3 × 2 system of equations above. Its coefficients

can be displayed in the 3 × 2 matrix

called the coefficient matrix for the system. If you

want to include the constants a a matrix, too, you

can add another column to get the 3 × 3 matrix

called the augmented matrix for the system. Frequently,

a vertical line is added to separate the coefficients

from the constants, as in

**Standard terminology and notation for matrices.**

When a single symbol is used to denote

an entire matrix, it’s usually a capital letter, like A

or B.

Usually m is used for the number of rows and

n for the number of columns. Such a matrix is

called an m × n matrix. When m = n we say the

matrix is a square matrix . When symbols are used

for the elements (that is, entries) of a matrix, they

are often doubly indexed (that is, subscripted), and

the indices indicate where the entry is located. For

instance, indicates the element in the 3rd
row

and 4th column. Note that the first index gives the

row number and the second index gives the column

number. When a generic row is needed, usually i is

used, and when a generic column is needed, usually

j is used. So is the element in the ith row
and

jth column.

**Special square matrices.** Square matrices were

mentioned above. There are some special square

matrices we’ll discuss in class including diagonal

matrices, scalar matrices, and the identity matrix.

A square matrix for which = 0 whenever i ≠
j

is called a diagonal matrix. The main diagonal of

a square matrix consists of the elements .
The

main diagonal is usually just called the diagonal.

Thus, a diagonal matrix only has nonzero entries on

its diagonal. If all the entries of a diagonal matrix

are the same, it’s called a scalar matrix. The most

important scalar matrix has all 1’s on its diagonal,

and that matrix is called an identity matrix, and

it’s usually denoted I. Thus, an identity matrix

has 1’s on its diagonal and 0’s elsewhere.

Some of the homework exercises talk about

upper-triangular matrices and lower-triangular matrices.

**Row and column vectors. **If a matrix has only

one row, that is, if m = 1, then we’ll call it a row

vector. Likewise, if it has only one column, that is,

if n = 1, then we’ll call it a column vector.

**Equality of matrices.** Two matrices are equal

when they have the same shape and all their corresponding

entries are equal.

**Addition and subtraction of matrices.** If two

matrices are the same shape, then you can add

them together to produce another matrix by adding

the corresponding elements together. Likewise you

can subtract one matrix from another if they have

the same shape. Examples 11 and 13 in the text

illustrate when these operations are useful .

**Scalar multiplication. **The word “scalar” in

this subject means “number,” and by that we usually

mean real number . Later in the course we’ll

take our field of scalars to be the field of complex

numbers or some other field. For now we’ll stick to

reals. We’ll use the standard notation R for the set

of all real numbers (that is, all positive numbers ,

negative numbers , and 0).

Scalar multiplication is an operation where you

can multiply a scalar and a matrix together. It’s

done by multiplying every entry in the matrix by

that scalar. For instance, if A is a matrix, then 2A

is found by doubling every entry in A. Note that

2A = A+A, just as you would expect. Also, when

A and B are two matrices , then A−B = A+(−1)B,

just as you would expect.

**Linear combinations.** In mathematics, you say

that a linear combination of things is a sum of multiples

of those things. So, for example, one linear

combination of the matrices A, B, and C is

2A+3B−4C. In other words, a linear combination

of matrices is found by the operations of addition,

subtraction, and scalar multiplication.

**The transpose of a matrix. **One more simple

operation we can discuss now is transposition. It’s

where you exchange rows and columns in a matrix.

For instance, if A is the matrix

,

then the transpose of A, denoted A^{T} is the
matrix

.

Note that if A is an m×n matrix, then its transpose

A^{T} is an n × m matrix. Also note that the transpose

of the transpose is the original matrix, that

is (A^{T} )^{T} = A. Also, note that transposition turns

row vectors into column vectors, and vice versa.

**Matrices in** Matlab. Last time we saw how

easy it was to enter into Matlab the coefficient

matrix A and constant column vector b of a system

of linear equations, and then compute the solution.

Let’s see what Matlab does for the 3 × 2 system

of equations above.

Let’s try Matlab on this system

Thus, (x, y, z) = (2, 1, 0) is a solution to the system

of equations. But it’s not the only solution.

Another is (x, y, z) = (3,−7, 11), and there are infinitely

more solutions to this indeterminate system.

On the other hand, the system

is inconstant, that is, it has no solutions. Matlab

gives

We’ll see later how Matlab can be used to help

you find all the solutions to a system.

Matlab also allows you to do create the identity

matrices of various sizes, do addition and subtraction

on matrices of the same shape, multiply a

scalar and a matrix, take a transpose of a matrix,

as well as many other operations, most of which

we’ll use throughout the course.

Prev | Next |