Subspaces in General Vector Spaces
Definition
 If a nonempty subset $W$ of a vector space $V$ is itself a vector space, we call $W$ a subspace in $V$.
Summary
 (Subspace Criteria) A subset $W$ of a vector space $V$ is a subspace if and only if
 The zero vector in $V$ is in $W$.
 For any vectors $A, B \in W$, the addition $A+B\in W$.
 For any vector $A\in W$ and a scalar $c$, the scalar multiplication $cA\in W$.
=solution
Problems

Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not.
(a) $S=\{f(x) \in V \mid f(0)=f(1)\}$.
(b) $T=\{f(x) \in V \mid f(0)=f(1)+3\}$. 
Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let
\[W=\{ f(x)\in V \mid f(x)=f(1x) \text{ for } x\in [0,1]\}\] be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$. 
Let $C[1,1]$ be the vector space over $\R$ of all realvalued continuous functions defined on the interval $[1, 1]$. Consider the subset $F$ of $C[1, 1]$ defined by
\[F=\{ f(x)\in C[1, 1] \mid f(0) \text{ is an integer}\}.\] Prove or disprove that $F$ is a subspace of $C[1, 1]$. 
Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient. Let $W$ be the following subset of $P_3$.
\[W=\{p(x) \in P_3 \mid p'(1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\] Here $p'(x)$ is the first derivative of $p(x)$ and $p^{\prime\prime}(x)$ is the second derivative of $p(x)$. Show that $W$ is a subspace of $P_3$ and find a basis for $W$. 
Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.
(a) The set $S$ consisting of all $n\times n$ symmetric matrices.
(b) The set $T$ consisting of all $n \times n$ skewsymmetric matrices.
(c) The set $U$ consisting of all $n\times n$ nonsingular matrices.  Let $V$ denote the vector space of all real $n\times n$ matrices, where $n$ is a positive integer. Determine whether the set $U$ of all $n\times n$ nilpotent matrices is a subspace of the vector space $V$ or not.
 Let $V$ be a vector space over a scalar field $K$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset
\[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } a_1+a_2+\cdots+a_k=0\}.\] So each element of $W$ is a linear combination of vectors $\mathbf{v}_1, \dots, \mathbf{v}_k$ such that the sum of the coefficients is zero. Prove that $W$ is a subspace of $V$. 
Suppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$ so that $S^{1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space.
(MIT) 
Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\] Let $U$ be the subset of $V$ defined by
\[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\] Prove that $U$ is a subspace of $V$.  Let $V$ be a vector space over a field $K$. If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset
\[W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}\] is a subspace of the vector space $V$.