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.

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

the entire expression should be subtracted from 30. to subtract the quantity, distribute the negative sign, which changes the signs on the terms. so, the –2 changes to positive and gets added to 30 instead of subtracted from it.

hope this ! ^w^

hdjd

step-by-step explanation: