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Problem 75: For the following subsets of $\R^n$ explain why they are or are not subspaces of $\R^n$

Problem 75: For the following subsets of $\R^n$ explain why they are or are not subspaces of $\R^n$ : $(a)$ The set of all linear combinations of two vectors $v, w \in \R^n$. $(b)$ The set of all vectors with first component equal to 2. $(c)$ The set of all vectors with first component equal to 0.

Solution:
$(a)$: Subspace
Closure under Addition:
$(c_1v+c_2w)+(d_1v+d_2w) = (c_1+d_1)v+(c_2+d_2)w$
Hence,W is closed under addition.
Closure under Scalar Multiplication:
Then, $ku = (kc_1+kc_2)$
which shows that $ku \in W$
Therefore W is a subspace of $\R^n$
$(b)$: Not a Subspace
Closure under Addition:
let $(2,4)$ and $(2,1)$ are vectors with first component equal to 2.
$(2,4)+(2,1) = (4,8) \notin W $
Hence,W is not closed under addition.
Therefore W is Not a subspace of $\R^n$
$(c)$: Subspace
Closure under Addition:
let $(0,a)$ and $(0,b)$ are vectors with first component equal to 0.
$(0,a)+(0,b)= (0,a+b) \in W$
Hence,W is closed under addition.
Closure under Scalar Multiplication:
Then, $k(0,a) = (0,ka)$
which shows that $ku \in W$
Therefore W is a subspace of $\R^n$

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