QUESTION 1 Vector Space Axioms Let V be a set on which two operations, called vector...

QUESTION 1

Vector Space Axioms

Let V be a set on which two operations, called vector addition and vector scalar multiplication, have been defined. If u and v are in V , the sum of u and v is denoted by u + v , and if k is a scalar, the scalar multiple of u is denoted by ku . If the following axioms satisfied for all u , v and w in V and for all scalars k and l , then V is called a vector space and its elements are 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 distinct entries and also combination of positive and negative numbers is a vector space if vector addition is defined to be matrix addition and vector scalar multiplication is defined to be matrix scalar multiplication.

QUESTION 2

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

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

(b) < u + v, w > = < u, w + v, w > (Additive 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 inner product space.

Task: Show that the set V of all 3×3 matrices with distinct entries and also combination of positive and negative numbers is a inner product space if vector addition is defined to be standard matrix addition and vector scalar multiplication is defined to be matrix scalar multiplication.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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