Example The basis vectors
are mutually perpendicular.
bf Remark: For any vector x, the unit vector with
direction x is 
Results:
a.) z is perpendicular to every vector 
b.) If z is perpendicular to each vector
then z is perpendicular
to their linear
span. If
, i = 1, . . . , k then 
.
c.) Mutually perpendicular vectors are linearly independent .
Definition: The projection of x on y
If Ly = 1 then 
Lemma: Let 
Then x - z is orthogonal to y.
Proof:
GRAM-SCHMIDT ORTHOGONALIZATION PROCESS:
Given linearly independent vectors
, there exists mutually
perpendicular vectors
with the same linear span. This may be constructed sequentially
by setting
We can use unit length vectors
instead of ;
Example Compute 
for
MATRICES
A matrix is a rectangular array of real numbers . An m × k
matrix has m rows and k
columns.
A, R, ∑ boldface letters, denote matrices.
For example an (m × 1) matrix has m rows and 1 column, it is an m dimensional
vector or
column matrix.
m × 1 matrix is a column vector,
1 × m matrix is a row vector.
Definition: Let 
be two (m × k) matrices. We say that the two
matrices
are equal 
I. e. two matrices are equal if
(i) their dimensionality is the same,
(ii) every corresponding element is the same.
MATRIX OPERATIONS
Let A, B be two (m × k) matrices.
Matrix Addition : C = A + B is an (m × k) matrix with elements
Scalar Multiplication: c arbitrary scalar
where
Matrix Substraction 
A − B = A + (−1)B = C where
Definition: Transpose of a matrix A = (
, i = 1, . . .
,m j = 1, . . . , k, A' is defined as a
(k × m) matrix with element 
j = 1, . . . , k i = 1, . . . , m
That is the transpose of a matrix A is obtained from A by inter changing the rows
and columns.
Example: 
Theorem: Let A, B, C be (m × k) matrices, c, d are scalars
(a) (A + B) + C = A + (B + C)
(b) A + B = B + A
(c) c(A + B) = cA + cB
(d) (c + d)A = cA + dA
(e) (A + B)' = A' + B'
(f) (cd)A = c(dA)
(g) (cA)' = cA'
Definition: If numbers of rows are equal to the numbers columns of a matrix A,
then it is
called square matrix .
Definition: Let A be a (k×k) square matrix. The matrix A is said to be symmetric
if A = A' .
That is A is symmetric , 
i, j = 1, . . . , k
Examples:
(i) Let I denotes the (m × m) matrix with 1-s in the main diagonal and zeros
elsewhere . Then
I is symmetric.
(ii)
Definition: I the (k × k) matrix is defined as the
identity matrix if it has ones only in the
main diagonal and zeros elsewhere.
Definition: Matrix Product Suppose that
is an (m × n) matrix.
is an
(n × k) matrix. Then 
, where C is and (m × k) matrix with elements
, i = 1, ...,m j = 1, ..., k, where
is the scalar product of ith row of
A with jth column
of B,
Example: 
Theorem: Properties of matrix multiplication A ,B,C defined
such that the indicated
products are defined and a scalar c is given
(a)c(AB) = (cA)B
(b)A(BC) = (AB)C
(c)A(B + C) = AB + AC
(d)(B + C)A = BA + CA
(e)(AB)' = B'A'
