11
Linear dependence and independence
Definition: A nite set S = cfw_x1 , x2 , . . . , xm of vectors in Rn is said to be linearly
dependent if there exist scalars (real numbers) c1 , c2 , . . . , cm , not all of which are 0, such
that c1 x1 + c2 x2 + . .
5
Homogeneous systems
Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is
one in which all the numbers on the right hand side are equal to 0:
a11 x1 + . . . + a1n xn = 0
.
.
.
.
.
.
am1 x1 + . . . + amn xn = 0
In matrix fo
4
Elementary matrices, continued
We have identied 3 types of row operations and their corresponding elementary matrices.
To repeat the recipe: These matrices are constructed by performing the given row operation
on the identity matrix:
1. To multiply rowj
2
Matrices and systems of linear equations
You have all seen systems of linear equations such as
3x + 4y = 5
2x y = 0.
(1)
This system can be solved easily: Multiply the 2nd equation by 4, and add the two resulting
equations to get 11x = 5 or x = 5/11. Su
3
Elementary row operations and their corresponding
matrices
As well see, any elementary row operation can be performed by multiplying the augmented
matrix (A|y) on the left by what well call an elementary matrix. Just so this doesnt
come as a total shock
1
Matrices and matrix algebra
1.1
Examples of matrices
Denition: A matrix is a rectangular array of numbers and/or variables. For instance
4 2
0 3 1
x 3
A = 5 1.2 0.7
3
4
6 27
is a matrix with 3 rows and 5 columns (a 3 5 matrix). The 15 entries of the m
7
Square matrices, inverses and related matters
Square matrices are the only matrices that can have inverses, and for this reason, they are
a bit special.
In a system of linear algebraic equations, if the number of equations equals the number of
unknowns,
6
The Inhomogeneous system Ax = y, y = 0
Definition: The system Ax = y is inhomogeneous if its not homogeneous.
Mathematicians love denitions like this! It means of course that the vector y is not the zero
vector. And this means that at least one of the e
8
Square matrices continued: Determinants
8.1
Introduction
Determinants give us important information about square matrices, and, as well soon see, are
essential for the computation of eigenvalues. You have seen determinants in your precalculus
courses. F
9
The derivative as a linear transformation
9.1
Redening the derivative
Matrices appear in many dierent contexts in mathematics, not just when we need to solve
a system of linear equations. An important instance is linear approximation. Recall from
your c
10
Subspaces
Now, we are ready to start the course. From this point on, the material will be new to most
of you. This means that most of you will not get it at rst. Youll have to read each
lecture a number of times before it starts making sense; fortunate