The Three Axioms: How to Test for a Subspace in R^n

📚 Quick Study Guide

To determine if a subset $W$ of $R^n$ is a subspace, you must verify the following three axioms:

• ➕ Closure under addition: If $\mathbf{u}$ and $\mathbf{v}$ are in $W$, then $\mathbf{u} + \mathbf{v}$ must also be in $W$.
• 🔢 Closure under scalar multiplication: If $\mathbf{u}$ is in $W$ and $c$ is any scalar, then $c\mathbf{u}$ must also be in $W$.
• 📍 Contains the zero vector: The zero vector, $\mathbf{0}$, must be in $W$.

Practice Quiz

1. Which of the following is NOT a requirement for a subset $W$ of $R^n$ to be a subspace?
   1. Contains the zero vector.
   2. Closed under addition.
   3. Closed under scalar multiplication.
   4. Contains the identity vector.
2. Let $W = \{(x, y) \in R^2 : x = y\}$. Is $W$ a subspace of $R^2$?
   1. No, because it doesn't contain the zero vector.
   2. No, because it's not closed under scalar multiplication.
   3. Yes, it satisfies all three axioms.
   4. No, because it's not closed under addition.
3. If $W$ is a subspace of $R^n$, and $\mathbf{u}, \mathbf{v} \in W$, which of the following must also be in $W$?
   1. $\mathbf{u} - \mathbf{v}$
   2. $\mathbf{u} \times \mathbf{v}$
   3. $\frac{\mathbf{u}}{\mathbf{v}}$
   4. $\mathbf{u} \cdot \mathbf{v}$
4. Let $W = \{(x, y) \in R^2 : x^2 + y^2 = 1\}$. Is $W$ a subspace of $R^2$?
   1. Yes, because it contains the zero vector.
   2. Yes, because it's closed under addition.
   3. No, because it doesn't contain the zero vector and is not closed under scalar multiplication.
   4. Yes, because it's closed under scalar multiplication.
5. Which of the following subsets of $R^3$ is a subspace?
   1. $\left\{(x, y, z) : x + y + z = 1\right\}$
   2. $\left\{(x, y, z) : x = y = z\right\}$
   3. $\left\{(x, y, z) : x^2 + y^2 + z^2 = 0\right\}$
   4. $\left\{(x, y, z) : xyz = 0\right\}$
6. If $W$ is a subspace, and $\mathbf{u} \in W$, which of the following is always true?
   1. $-\mathbf{u} \notin W$
   2. $-\mathbf{u} \in W$
   3. $\mathbf{u} = \mathbf{0}$
   4. $\mathbf{u}$ is not a vector.
7. Let $W = \{(x, y, z) \in R^3 : z = 0\}$. Is $W$ a subspace of $R^3$?
   1. No, because it does not contain the zero vector.
   2. No, because it's not closed under scalar multiplication.
   3. Yes, it satisfies all three axioms.
   4. No, because it's not closed under addition.