Subspaces in General Vector Spaces

Subspaces in General Vector Spaces

Definition

  1. If a nonempty subset $W$ of a vector space $V$ is itself a vector space, we call $W$ a subspace in $V$.
Summary

  1. (Subspace Criteria) A subset $W$ of a vector space $V$ is a subspace if and only if
    1. The zero vector in $V$ is in $W$.
    2. For any vectors $A, B \in W$, the addition $A+B\in W$.
    3. For any vector $A\in W$ and a scalar $c$, the scalar multiplication $cA\in W$.

=solution

Problems

  1. 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) $T=\{f(x) \in V \mid f(0)=f(1)+3\}$.

  2. Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let
    \[W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}\] be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.

  3. Let $C[-1,1]$ be the vector space over $\R$ of all real-valued 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 $C[-1, 1]$.

  4. 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 $W$.

  5. 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 $n \times n$ skew-symmetric matrices.
    (c) The set $U$ consisting of all $n\times n$ nonsingular matrices.

  6. 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.

  7. 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 $V$.

  8. 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)

  9. 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 $V$.

  10. 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 $V$.