OOPS I MESSED UP THE TITLE CAN ANY1 TELL ME HOW I CAN CHANGE IT?

Let $\displaystyle V = R^3$, and consider the following subsets of V.

$\displaystyle U_1 = \{ t \begin{pmatrix}

1\\

0\\

0\end{pmatrix} : t \in R \}$

$\displaystyle U_2 = \{ t \begin{pmatrix}

0\\

1\\

0\end{pmatrix} : t \in R \}$

$\displaystyle U_3 = \{t \begin{pmatrix}

1\\

1\\

1\end{pmatrix} : t \in R \}$

$\displaystyle U_4 = \{ \begin{pmatrix}

r\\

s\\

0\end{pmatrix} : r,s \in R \}$

$\displaystyle U_5 = \{ \begin{pmatrix}

r\\

s\\

1\end{pmatrix} : r,s \in R \}$

$\displaystyle U_6 = \{ \begin{pmatrix}

r\\

s\\

0\end{pmatrix} : r,s,t \in R \text{ and } r+s+t=1 \}$

$\displaystyle U_7 = U_1 \cap U_2$

$\displaystyle U_8 = U_1 \cup U_2$

a) For each of the 8 sets sat whether it is a subspace, and briefly explain your answer.

b) For each of the 8 sets, classify it as one of

i. A line passing through the origin.

ii. A line not passing through the origin.

iii. A plane passing through the origin.

iv. A plane not passing through the origin.

v. none of the above

c) What do you conclude about the subspaces of $\displaystyle R^3$?