Examples of Non-Subspaces with Detailed Explanations and Counterexamples

📚 Quick Study Guide

• 🔢 A subspace must contain the zero vector. If a set doesn't, it's not a subspace.
• ➕ Subspaces must be closed under addition. If $\mathbf{u}$ and $\mathbf{v}$ are in the subspace, then $\mathbf{u} + \mathbf{v}$ must also be in the subspace.
• multiplied by a scalar. If $\mathbf{u}$ is in the subspace, then $c\mathbf{u}$ must also be in the subspace for any scalar $c$.
• 🛑 To prove a set is *not* a subspace, you only need to find one counterexample that violates one of these conditions.

Practice Quiz

1. Which of the following sets is NOT a subspace of $\mathbb{R}^2$?

   1. {(x, y) | x + y = 0}
   2. {(x, y) | x = y}
   3. {(x, y) | xy = 0}
   4. {(x, y) | x - y = 0}

2. Which of the following sets is NOT a subspace of $\mathbb{R}^3$?

   1. {(x, y, z) | x + y + z = 0}
   2. {(x, y, z) | x = y = z}
   3. {(x, y, z) | x^2 + y^2 + z^2 = 1}
   4. {(x, y, z) | x - y = 0}

3. Consider the set of all $2 \times 2$ matrices with determinant equal to 0. Is this a subspace of the vector space of all $2 \times 2$ matrices?

   1. Yes, it is closed under addition and scalar multiplication.
   2. Yes, it contains the zero matrix.
   3. No, it is not closed under addition.
   4. Yes, all of the above conditions are met.

4. Let $V$ be the vector space of all functions $f: \mathbb{R} \to \mathbb{R}$. Which of the following is NOT a subspace of $V$?

   1. The set of all differentiable functions.
   2. The set of all continuous functions.
   3. The set of all functions $f$ such that $f(0) = 1$.
   4. The set of all polynomial functions.

5. Which of the following is NOT a subspace of the vector space of all polynomials?

   1. The set of all polynomials of degree at most 5.
   2. The set of all polynomials with only even powers.
   3. The set of all polynomials with a root at $x = 2$.
   4. The set of all polynomials of degree exactly 5.

6. Let $S = \{(x, y) \in \mathbb{R}^2 : x^2 = y^2\}$. Is $S$ a subspace of $\mathbb{R}^2$?

   1. Yes, it satisfies all the subspace conditions.
   2. Yes, it contains the zero vector.
   3. No, it is not closed under scalar multiplication.
   4. No, it is not closed under addition.

7. Which of the following sets of vectors in $\mathbb{R}^2$ does NOT form a subspace?

   1. The set of all vectors along the line y = 2x.
   2. The set containing only the zero vector.
   3. The set of all vectors (x, y) where x is a non-negative integer.
   4. The set of all vectors along the x-axis.