Question

Determine whether each of the following subsets of R3 is a subspace of R3:

a) {(x,y,z) is an element of R3 : x = 0}
b) {(x,y,z) is an element of R3 : x + y = 0}
c) {(x,y,z) is an element of R3 : xz = 0}
d) {(x,y,z) is an element of R3 : y > or = 0}
d) {(x,y,z) is an element of R3 : x = y = z}

Answer 1

What have you gotten so far?

Answer 2

I don't understand subsets and subspace too well. If you can show me how to do a) at least, that would be great. @cruffo

Answer 3

There are three things that have to be satisfied in order for a subset to be a subspace. Can you state any of the three conditions?

Answer 4

The definition that I have in my notes says:

DEFINITION: Suppose V is a vector space over R

A subset \[W \subseteq V\] is called a (vector) subspace if:

1. w1, w2 \[\in W\] --> w1 + w2 \[\in W\]

2. w \[\in W\] and c \[\in R\] --> cw \[\in W\]

@cruffo

Answer 5

Good. The three conditions are
(i) 0 is in the set
(ii) if x and y are in the set, then x + y is also in the set
(iii) if c is a scalar and x is in the set, then cx is also in the set

Answer 6

@infinitylove

a) {(x,y,z) is an element of R3 : x = 0} All the vectors of this set are of the form (0,y,z) where y and z are real numbers. Is (0,0,0) in the set??

Answer 7

Yes. So that satisfies the first condition, correct?

Answer 8

@cruffo

Answer 9

Yep. Good. Now for the second: Take any two vectors of the form (0,x,y) and add them. Do you get another vector of the same form?

(0,a,b) + (0,c,d) = ??

Answer 10

The answer to that would be (0, a+c, b+d). Can you explain how that helps for the second condition? @cruffo

Answer 11

For (a) you just have to make sure you end up with a vector of the form (0, real #, real #).
a, b,c,d are real # to start with.
so a+c and b+d are real #.

Answer 12

Okay. I understand. So that would satisfy condition 2, correct? @cruffo

Answer 13

Right! Now for (iii) if you take a number and multiply it to a vector of the form (0,x,y) do you get a vector of the same form?

c(0,x,y) = ???

Answer 14

YES! haha, so then we know that this subset of R3 is also a subspace of R3? @cruffo

Answer 15

You got it ; )

Answer 16

Thanks a lot! And its the same process for all the others? @cruffo

Answer 17

Pretty much. You just have to watch out for the specification of the form of the vectors:

b) {(x,y,z) is an element of R3 : x + y = 0}
x + y =0 means y = -x
The vectors of this set have the form (x,-x,z)

c) {(x,y,z) is an element of R3 : xz = 0}
If xz = 0, then either x = 0, z = 0 or they are both 0.

That kind of thing.

Answer 18

I don't think (d) will be a subspace. Looks like it will fail condition (iii)

Answer 19

Okay. I will keep this all in mind when I continuously work on this problem tomorrow. Thanks so much for all your help and I will definitely contact you if I need any more help. THANKS! @cruffo

Answer 20

Cheers!