VECTORS in R^{n} and VECTOR EQUATIONS
 n = 2:

,
a 2vector.
Addition
,
scalar multiplication
,
zero vector
.
Geometry and parallelogram rule
 n = 3:

,
a 3vector.
Addition
,
scalar multiplication
,
zero vector
.
 General n:

,
an nvector.
Addition
,
scalar multiplication
,
zero vector
.
Algebraic properties:
for all
,
and scalars c, d
 1.
 u + v = v + u
 2.
 ( u + v ) + w = u + ( v + w )
 3.
 u + 0 = u
 4.
 u + (u) = u  u= 0
 5.
 c(u + v) = cu + cv
 6.
 (c+d)u = cu + du
 7.
 c(du) = (cd)u
 8.
 1u = u
 Linear combinations:
 linear combination of
,
using
,
is

is the set of all possible
linear combinations
Geometry: For
,
is a line,
is (usually) a plane.
Vector equations:
iff the linear system
is consistent.
The MATRIX EQUATION
 MatrixVector Multiplication:
 If A is an
matrix and
,
 Linear Systems:
 These representations all have the same solution set:

,
 vector equation
 linear system with augmented matrix
 Existence of solutions:

has a
solution iff
is a linear combination of
.
 Computation of :

 Identity Matrix:

 I has ones on the diagonal and zeros elsewhere.
 I_{n} is an
identity matrix.

for all
.
Alan C Genz
20000706