Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.4 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

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.

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,

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,

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

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.

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,

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

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

Exercise: Prove the triangle inequality