Determining if Something is a Subspace Which of the following subsets $U\subset \mathbb{R}^{n}$ is a subspace?
a) $U = \{x \in \mathbb{R}^n\,|\, x_1=\dotsb=x_n\}$
b) $U = \{x \in \mathbb{R}^n\,|\, x^2_1=x^2_2\}$
c) $U = \{x \in \mathbb{R}^n\,|\, x_1=1\}$
d)  All of a, b, c are subspaces
e) None of a, b, c are subspaces
I know the properties of a subspace, but I am confused on how you apply that here. I don't understand the actual procedure to test each one. Thanks
 A: $b)$ isn't a subset because the vector $x_1=(1,-1,...,0)$ and $x_2=(1,1,...,0)$ belongs to $U$ but $x_1+x_2=(2,0,...,0)$ doesn't belong to $U$.
$c)$ isn't a subset because the vector $(0,0,...,0)$ can't belong to $U$.
A: To show $U$ is a subspace, you need to show $(1)$ that $\vec0\in U$, $(2)$ if $x,y\in U$ then $x+y\in U$, $(3)$ if $x\in U$ and $a\in\Bbb R$, then $ax\in U$.
Let's look at (a). Clearly $\vec0=(0,\dots,0)\in U$. If $x,y\in U$ then we can write $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ where $x_1=\dots=x_n$ and $y_1=\dots=y_n$. Then, $x+y=(x_1+y_1,\dots,x_n+y_n)$, and you can see that we must have $x_1+y_1=\dots=x_n+y_n$, so $x+y\in U$. Similarly, you can show $ax\in U$ if $a\in\Bbb R$.
What about (b)? Looking at condition $(2)$, if $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ are in $U$, so $x_1^2=x_2^2$ and $y_1^2=y_2^2$, then is it always true that $(x_1+y_1)^2=(x_2+y_2)^2$? You should be able to find an easy example where this fails.
Now we look at (c). It shouldn't be difficult to see that condition $(1)$ fails here.
