Linear Algebra
Notes from Kolman and Hill, 9th edition:
An mxn
linear system
Ax = b is a system of m linear equations in n unknowns x
i
, i=1,.
..,n.
If the linear system has a solution it is
consistent,
otherwise it is
inconsistent
.
Ax = 0 is a
homogeneous
linear system.
The homogeneous linear system always has the
trivial solution
x = 0.
The linear systems Ax = b and Cx = d are
equivalent
, if they both have exactly
the same solutions.
Def 1.1: An mxn
matrix
A is a rectangular array of mn real or complex numbers
arranged in m horizontal rows and n vertical columns.
Def 1.2: Two mxn matrices A=[a
ij
] and B=[b
ij
] are
equal
, if they agree entry by
entry.
Def 1.3: The mxn matrices A and B are
added
entry by entry.
Def 1.4: If A=[a
ij
] and r is a real number then the
scalar multiple
of A is the
matrix rA=[ra
ij
].
If A
1
, A
2
, .
.., A
k
are mxn matrices and c
1
, c
2
, .
.., c
k
are real numbers then an
expression of the form
c
1
A
1
+ c
2
A
2
+ .
.. + c
k
A
k
is a
linear combination
of the A's with coefficients c
1
, c
2
, .
.., c
k
.
Def 1.5: The
transpose
of the mxn matrix A=[a
ij
] is the nxm matrix A
T
=[a
ji
].
Def 1.6: The
dot product
or inner product of the n-vectors a=[a
i
] and b=[b
i
] is ab
= a
1
b
1
+ a
2
b
2
+ .
.. + a
n
b
n
.
Def 1.7: If A=[a
ij
] is an mxp matrix and B=[b
ij
] a pxn matrix they can be
multiplied
and the ij entry of the mxn C = AB:
c
ij
= dot product of the (ith row of A)
T
with (jth column of B).
Note: Ax = b is consistent if and only if b can be expressed as a linear
combination of the columns of A with coefficients x
i
.
Theorem 1.1
Let A, B, and C be mxn matrices, then
(a) A + B = B + A
(b) A + (B + C) = (A + B) + C
(c) there is a unique mxn matrix O such that for any mxn matrix A
A + O = A
(d) for each mxn matrix A, there is aunique mxn matrix D such that
A + D = O
D = -A is the negative of A.
Theorem 1.2
Let A, B, and C be matrices of the appropriate sizes, then
(a) A(BC) = (AB)C
(b) (A + B)C = AC + BC
(c) C(A + B) = CA + CB
Theorem 1.3
Let r, s be real numbers and A, B matrices of the appropriate sizes,
then
(a) r(sA) = (rs)A