Test Questions for Proving Subspaces and Non-Subspaces with Answers

📚 Quick Study Guide

• ➕ A subset $W$ of a vector space $V$ is a subspace if it satisfies three conditions:
• 🔒 Closure under addition: If $u$ and $v$ are in $W$, then $u + v$ is in $W$.
• 🔢 Closure under scalar multiplication: If $u$ is in $W$ and $c$ is a scalar, then $cu$ is in $W$.
• 🎯 Non-emptiness: $W$ must contain the zero vector $0$.
• 💡 To prove a subset is NOT a subspace, you only need to show that ONE of these conditions fails. Common strategies involve showing that the zero vector is not in the set, or demonstrating that the set isn't closed under addition or scalar multiplication.
• 🚫 A set that does not contain the zero vector cannot be a subspace.
• 🧪 Example of proving closure under scalar multiplication: Let $W = \{(x, x) : x \in \mathbb{R}\}$. If $(x, x) \in W$, then $c(x, x) = (cx, cx)$. Since $cx$ is a real number, $(cx, cx) \in W$. Therefore, $W$ is closed under 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. It must contain the zero vector.
   2. It must be closed under addition.
   3. It must be closed under scalar multiplication.
   4. It must be non-empty.
2. Which of the following sets is a subspace of $\mathbb{R}^2$?
   1. $\left\{ (x, y) : x^2 + y^2 = 1 \right\}$
   2. $\left\{ (x, y) : x = y \right\}$
   3. $\left\{ (x, y) : x = 1 \right\}$
   4. $\left\{ (x, y) : xy = 0 \right\}$
3. Let $V = \mathbb{R}^2$. Is $W = \{(x, y) : x \geq 0, y \geq 0\}$ a subspace of $V$?
   1. Yes
   2. No, because it is not closed under scalar multiplication.
   3. No, because it does not contain the zero vector.
   4. No, because it is not closed under addition.
4. Which of the following sets is NOT a subspace of $\mathbb{R}^3$?
   1. $\left\{ (x, y, z) : x + y + z = 0 \right\}$
   2. $\left\{ (x, y, z) : x = y = z \right\}$
   3. $\left\{ (x, y, z) : x = 0 \right\}$
   4. $\left\{ (x, y, z) : x = 1 \right\}$
5. Let $V$ be a vector space. If $W = \{0\}$, is $W$ a subspace of $V$?
   1. Yes
   2. No, because it is too small.
   3. No, because it does not contain enough vectors.
   4. It depends on the vector space $V$.
6. Consider the set $W = \{(a, b, a+b) : a, b \in \mathbb{R}\}$. Is $W$ a subspace of $\mathbb{R}^3$?
   1. Yes
   2. No, because it is not closed under addition.
   3. No, because it is not closed under scalar multiplication.
   4. No, because it does not contain the zero vector.
7. Is the set of all $2 \times 2$ matrices with determinant equal to 0 a subspace of the vector space of all $2 \times 2$ matrices?
   1. Yes
   2. No, because it is not closed under addition.
   3. No, because it is not closed under scalar multiplication.
   4. No, because it does not contain the zero matrix.