Question

Given subspaces H and K of a vector space V, the sum of H and K, written as H+K, is the set of all vectors in V that can be written as the sum of two vectors, one in H and the other in K; that is, H+K={w:w=u+v for some u in H and some v in K}
a. Show that H+K is a subspace of V.
b. Show that H is a subspace of H+K and K is a subspace of H+K.

Answer 1

Answer:

a. H + K = {u + v}

b. See explanation below

Step-by-step explanation:

Given

H+K={w:w=u+v for some u in H and some v in K}

a.

From the (given) above,

We have that

0 = v1 + 0.u

v1 represents the zero vector in K

We also have that

0 = v + u1

u1 represents the zero vector in H

Up this point, we've shown that both H and K have vector factors, V

We'll use the available data to solve for the zero vector of H + K

This is given by

u1 + v1 + u2 + v2 ---- Rearrange

u1 + u2 + v1 + v2 ---- Let u3 = u1 + u2 and v3 = v1 + v2.

So, we have

u3 + v3 as the zero vector of H + K

H + K = {u + v}

So, H + K is a subspace of V

b

Since H, K are subspace of V, then they are

1. Both close under scalar multiplication

2. Both closed under vector addition

1 and 2 gives

c(u + v) where c is a constant

= c.u + c.v

We can then conclude that H is a subspace of H+K and K is a subspace of H+K because H + K is closed under scalar multiplication and vector addition