Which of the following subsets of R3 are subspacesExplain a

Which of the following subsets of R3 are subspaces.Explain.

a) {(x, y, z) | x 0, y 0, z 0}

b) {(x, y, z) | (all) y, x = z}

c) {(x, y, z) | x 2 + y 2 + z 2 1}.

d) Is the set H of all matrices of the form [(a, 0)T ,(b, d) T ] a subspace of the space of all 2x2 matrices with the usual matrix addition and scalar multiplication?

Solution

a)

No.

It is not closed under scalar multiplication

Consider the vector in this set

(1,-1,1)

Multiplying this by -1 gives

(-1,1,-1) which is not in this set

HEnce not a subpsace

b)

Yes.

1. Let, (x,y,z),(u,v,w)

(x,y,z)+(u,v,w)=(x+u,y+v,z+w)

x+u=z+w

and y+z is in R

So the set is closed under addition

2. Let, (x,y,z) be in the set and c be a scalar from R

c(x,y,z)=(cx,cy,cz)
cx=cz

cy is in R

HEnce closed under scalar multiplication

Hence a subspace

c)

Not a subspace

Consider an elemint in this set

(1,0,0)

MUltiply this by scalar: 0.5 gives the element

(1/2,0,0)

1/2^2+0+0=1/4<1

HEnce the element (1/2,0,0) is nnot in the set

So the set is not a subspace as it is not closed under scalar multiplicatin

d)

Yes.

The defining property of these matrices is that entry in (2,1) is equal to 0

Let, A , B be two matrice in H

(A+B)_{2,1}=A_{2,1}+B_{2,1}=0

SO, A+B is in H

Let, A be in H and c be a scalar

(cA)_{2,1}=cA_{2,1}=0

HEnce, cA is in H

SO, H is closed under matrix addition and scalar multiplication

Hence, H is a subspace