Solved Examples: Proving Subspaces in Abstract Vector Spaces

📚 Quick Study Guide

• 🔍 Definition of a Subspace: A subset $W$ of a vector space $V$ is a subspace if it satisfies three conditions:
  • Contains the zero vector of $V$.
  • Closed under vector addition: If $u, v \in W$, then $u + v \in W$.
  • Closed under scalar multiplication: If $u \in W$ and $c$ is a scalar, then $cu \in W$.
• 💡 Zero Vector Test: Always check if the zero vector of the parent vector space is in the subset. If not, the subset is not a subspace.
• 📝 Closure Under Addition: To prove closure under addition, take two arbitrary vectors in the subset and show that their sum is also in the subset.
• ➗ Closure Under Scalar Multiplication: To prove closure under scalar multiplication, take an arbitrary vector in the subset and an arbitrary scalar, then show that the scalar multiple of the vector is also in the subset.
• 📌 Common Mistakes: Forgetting to verify all three conditions or making incorrect assumptions about the vectors in the subset.
• 🧮 Useful Tip: If a subset fails any of the three conditions, it is not a subspace.

Practice Quiz

1. Question 1: Which of the following is NOT a requirement for a subset $W$ of a vector space $V$ to be a subspace?
   1. Contains the zero vector of $V$.
   2. Closed under vector addition.
   3. Closed under scalar multiplication.
   4. Contains the identity element for multiplication.
2. Question 2: Let $V = \mathbb{R}^2$. Is $W = \{(x, y) \in V : x = y\}$ 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.
3. Question 3: Let $V = \mathbb{R}^2$. Is $W = \{(x, y) \in V : x = 1\}$ a subspace of $V$?
   1. Yes
   2. No, because it does not contain the zero vector.
   3. No, because it is closed under addition.
   4. No, because it is closed under scalar multiplication.
4. Question 4: Let $V = P_2(\mathbb{R})$, the vector space of polynomials of degree at most 2. Is $W = \{p(x) \in V : p(0) = 0\}$ 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.
5. Question 5: Let $V = M_{2x2}(\mathbb{R})$, the vector space of $2 \times 2$ matrices. Is $W = \{A \in V : A = A^T\}$ (symmetric matrices) 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.
6. Question 6: Let $V = \mathbb{R}^3$. Is $W = \{(x, y, z) \in V : x^2 + y^2 + z^2 = 1\}$ 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.
7. Question 7: Let $V = F(\mathbb{R}, \mathbb{R})$ be the vector space of all functions from $\mathbb{R}$ to $\mathbb{R}$. Is $W = \{f \in V : f(1) = 1\}$ 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.