What is a Subspace? Definition & Examples in Linear Algebra

📚 Quick Study Guide

• 🔍 Definition: A subspace $W$ of a vector space $V$ is a subset of $V$ that is itself a vector space under the same operations of addition and scalar multiplication defined on $V$.
• ➕ Closure under Addition: If $\mathbf{u}$ and $\mathbf{v}$ are in $W$, then $\mathbf{u} + \mathbf{v}$ is also in $W$.
• scale Closure under Scalar Multiplication: If $\mathbf{u}$ is in $W$ and $c$ is a scalar, then $c\mathbf{u}$ is also in $W$.
• 🧮 Zero Vector: Every subspace must contain the zero vector, $\mathbf{0}$. This is often the easiest way to disprove that a set is a subspace.
• 📝 Common Examples: The set containing only the zero vector $\{\mathbf{0}\}$ is a subspace, as is the entire vector space $V$ itself. These are known as the trivial subspaces.
• 💡 How to Check: To verify that a subset $W$ is a subspace of $V$, you must verify that $W$ is non-empty (often done by showing $\mathbf{0} \in W$), and that $W$ is closed under vector addition and scalar multiplication.

Practice Quiz

1. Which of the following is NOT a requirement for a subset $W$ of a vector space $V$ to be a subspace?

   1. $W$ contains the zero vector.
   2. $W$ is closed under addition.
   3. $W$ is closed under scalar multiplication.
   4. $W$ is non-empty.

2. Let $V = \mathbb{R}^2$. Which of the following subsets of $V$ is a subspace?

   1. $\lbrace (x, y) \mid x + y = 1 \rbrace$
   2. $\lbrace (x, y) \mid x = y \rbrace$
   3. $\lbrace (x, y) \mid x^2 + y^2 = 1 \rbrace$
   4. $\lbrace (x, y) \mid xy = 0 \rbrace$

3. Let $V = \mathbb{R}^3$. Is $W = \lbrace (x, y, z) \mid x - y + 2z = 0 \rbrace$ a subspace of $V$?

   1. Yes
   2. No, because it does not contain the zero vector.
   3. No, because it is not closed under addition.
   4. No, because it is not closed under scalar multiplication.

4. Which of the following is always a subspace of any vector space $V$?

   1. The empty set.
   2. The set containing only the zero vector $\{\mathbf{0}\}$.
   3. Any set containing only one non-zero vector.
   4. Any finite set of vectors.

5. Let $V$ be the vector space of all $2 \times 2$ matrices. Which of the following is a subspace of $V$?

   1. The set of all invertible $2 \times 2$ matrices.
   2. The set of all singular $2 \times 2$ matrices.
   3. The set of all $2 \times 2$ matrices with determinant equal to 1.
   4. The set of all $2 \times 2$ matrices with trace equal to 0.

6. If $U$ and $W$ are subspaces of a vector space $V$, which of the following is necessarily also a subspace of $V$?

   1. $U \cup W$ (the union of $U$ and $W$).
   2. $U \setminus W$ (the set difference of $U$ and $W$).
   3. $U \cap W$ (the intersection of $U$ and $W$).
   4. The complement of $U$ in $V$.

7. Let $P_n$ be the vector space of all polynomials of degree at most $n$. Is the set of all polynomials in $P_n$ with a constant term of 0 a subspace?

   1. Yes
   2. No, because it does not contain the zero polynomial.
   3. No, because it is not closed under addition.
   4. No, because it is not closed under scalar multiplication.