Determine whether the following subsets are subspaces

[physicsNYC]

Homework Statement

H = {(x,y,z) $$\in$$ R^3 | x + y^2 + z = 0} $$\subseteq$$ R^3

T = {A $$\in$$ M_2,2 | A^T = A} $$\subseteq$$ M_2,2

The Attempt at a Solution

Our lecturer wasn't quite clear about how to go about this.

He talked out closed under addition and multiplication but that's about it.

Help would be greatly appreciated

 

[Mark44]

Homework Statement

H = {(x,y,z) $$\in$$ R^3 | x + y^2 + z = 0} $$\subseteq$$ R^3

T = {A $$\in$$ M_2,2 | A^T = A} $$\subseteq$$ M_2,2

The Attempt at a Solution

Our lecturer wasn't quite clear about how to go about this.

He talked out closed under addition and multiplication but that's about it.

Help would be greatly appreciated

This is pretty basic stuff in linear algebra, so I'm surprised that your lecturer wasn't sure how to do this. What he said, though, is pretty much what you need to do.

For your first problem, here's what you need to do:

1. Show that the zero vector is in H.
2. Show that if h₁ and h₂ are any two vectors in set H, then h₁ + h₂ is also in H. (Closure under vector addition)
3. Show that if h₁ is a vector in H and c is any real number, then c*h₁ is in H. (Closure under scalar multiplication) Note that the first step can be accomplished by using c = 0.

For your second problem use matrices instead of vectors, but the steps are essentially the same.