Question

let v be a vector space over f. show that the sum of two inner products on v is an inner product on v . show that a positive multiple of an inner product is also an inner product.

Answer 1

To show that the sum of two inner products on vector space V is an inner product on V, and that a positive multiple of an inner product is also an inner product, we need to prove that they satisfy the properties of an inner product.

Step-by-step explanation:

To show that the sum of two inner products on vector space V is an inner product on V, we need to prove that it satisfies the properties of an inner product. Let's assume that <u,v> and <w,x> are two inner products on V, where u, v, w, and x are vectors in V, and c is a scalar. The sum of these two inner products, denoted as <u,v>+<w,x>, can be defined as:

<u,v>+<w,x> = <cu+dw, cv+dx>

In order to prove that this is an inner product, we need to show that it satisfies the following properties:

1. Linearity in the first argument: <au+v, w> = a<u,w> + <v,w>
2. Conjugate symmetry: <u,v> = <v,u>
3. Positive-definiteness: <v,v> > 0 for all nonzero v in V

To prove that a positive multiple of an inner product is also an inner product, we can use a similar approach. Let <u,v> be the inner product on V, where u and v are vectors in V. Multiplying this inner product by a positive scalar c, denoted as c<u,v>, can be defined as:

c<u,v> = <cu, v>

We also need to show that this satisfies the properties of an inner product as listed above.