1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.4
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Definition of projection of vector: Consider two vector a and b and is θ angle between these vector.
Scalar projection of vector b along the vector a = b cosθ
Vector projection of vector b along the direction of vector a can be written as, =
If vector a is of unit norm, then, = 1.

3. Gram-Schmidt orthonormalisation method:
A set of linealry independent basis vectors with each of
unit norm and which are pairwise orthogonal are known to
form ortho-normal basis.
E.g. Two basis and will form orthonormal basis
If,

4. Gram-Schmidt orthonormalization method is a procedure
to convert a given linearly independent basis to orthonormal
basis.
Let are the linearly independent basis vectors
which we want to orthonormalize.
Step 1: Rescale the first vector by its norm. We get 1st vector
of orthonormalize set
Note that,

5. Step 2: Substract from 2nd vector its projection along the
1st vector leaving behind only the part perpendicular to 1st.
So we get
Note that above vector is orthogonal to
Dividing by its norm we will get 2nd basis vector
of orthonormalized set which is orthogonal to 1st and also of
unit norm. =1

6. Consider
Above vector is orthogonal to both |1> and |2> . Dividing
its norm we will get the third vector of orthonormal
basis set.
The above procedure can be extended to other vectors
of linearly independent basis to form orthonormal basis.

7. Schwarz Inequality: It is the generalization for the
angle between two vector for vector space.
The dot product of two vectors cannot exceed the product
of their length.
Schwartz Inequality is,
Proof:
We define,

8. To prove Schwarz Equality we made use of fact that
We find
(1)
Now
(2)
Which is real quantity. It means above equation gives

9. Also
Which gives us
------(3)
Using (2) and (3) in (1), we get
Which implies
Which is Schwarz Inequality.

10. Exercise: Prove the triangle inequality