QUESTION 1

Vector Space Axioms

Let V be a set on which two operations, called vector additionand vector scalar multiplication, have been defined. If u and v arein V , the sum of u and v is denoted by u + v , and if k is ascalar, the scalar multiple of u is denoted by ku . If thefollowing axioms satisfied for all u , v and w in V and for allscalars k and l , then V is called a vector space and its elementsare called vectors.

1) u + v is in V

2) u + v = v + u

3) (u + v) + w = u + (v + w)

4) 0 + v = v

5) v + (?v) = 0

6) ku is in V

7) k(u + v) = ku + kv

8) (k + l)u = ku + lu

9) k(lu) = (kl)(u)

10) 1v = v

Task: Show that the set V of all 3×3 matrices with distinctentries and also combination of positive and negative numbers is avector space if vector addition is defined to be matrix additionand vector scalar multiplication is defined to be matrix scalarmultiplication.

QUESTION 2

Suppose u, v, and w are all vectors in a vector space V and c isany scalar. An inner product on the vector space V is a functionthat associates with each pair of vectors in V, say u and v, a realnumber denoted by u, v that satisfies the following axioms.

(a) < u, v > = < v, u > (Symmetry axiom)

(b) < u + v, w > = < u, w + v, w > (Additiveaxiom)

(c) < cu, v > = < c u, v > (Homogeneity axiom)

(d) < u, u > ? 0 and < u, u > = 0 if and only if u =0 (Positivity axiom)

A vector space along with an inner product is called an innerproduct space.

Task: Show that the set V of all 3×3 matrices with distinctentries and also combination of positive and negative numbers is ainner product space if vector addition is defined to be standardmatrix addition and vector scalar multiplication is defined to bematrix scalar multiplication.