Test Your Knowledge: Basis for a Vector Space Exam Questions

📚 Quick Study Guide

• ➕ Vector Space Definition: A set $V$ with two operations (addition and scalar multiplication) satisfying certain axioms.
• 🔢 Axioms of Vector Addition:
  • Closure: $\mathbf{u} + \mathbf{v} \in V$ for all $\mathbf{u}, \mathbf{v} \in V$
  • Commutativity: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ for all $\mathbf{u}, \mathbf{v} \in V$
  • Associativity: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$ for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$
  • Additive Identity: There exists a $\mathbf{0} \in V$ such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$ for all $\mathbf{u} \in V$
  • Additive Inverse: For every $\mathbf{u} \in V$, there exists a $-\mathbf{u} \in V$ such that $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$
• ➗ Axioms of Scalar Multiplication:
  • Closure: $c\mathbf{u} \in V$ for all $c \in \mathbb{R}$ and $\mathbf{u} \in V$
  • Distributivity (scalar): $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$ for all $c \in \mathbb{R}$ and $\mathbf{u}, \mathbf{v} \in V$
  • Distributivity (vector): $(c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}$ for all $c, d \in \mathbb{R}$ and $\mathbf{u} \in V$
  • Associativity: $c(d\mathbf{u}) = (cd)\mathbf{u}$ for all $c, d \in \mathbb{R}$ and $\mathbf{u} \in V$
  • Multiplicative Identity: $1\mathbf{u} = \mathbf{u}$ for all $\mathbf{u} \in V$
• 📏 Subspace: A subset $W$ of $V$ is a subspace if it is closed under vector addition and scalar multiplication.
• 💡 Linear Combination: A vector $\mathbf{v}$ is a linear combination of vectors $\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n$ if $\mathbf{v} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + ... + c_n\mathbf{v}_n$ for scalars $c_1, c_2, ..., c_n$.
• 🔑 Span: The span of a set of vectors is the set of all linear combinations of those vectors.
• 📈 Linear Independence: Vectors $\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n$ are linearly independent if $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + ... + c_n\mathbf{v}_n = \mathbf{0}$ implies $c_1 = c_2 = ... = c_n = 0$.
• 🎯 Basis: A basis for a vector space $V$ is a set of linearly independent vectors that span $V$.
• 📐 Dimension: The dimension of a vector space $V$ is the number of vectors in a basis for $V$.

Practice Quiz

1. Which of the following is NOT a required axiom for a set to be considered a vector space?
   1. (A) Closure under vector addition
   2. (B) Existence of a multiplicative inverse
   3. (C) Existence of an additive identity
   4. (D) Closure under scalar multiplication
2. Let $V = \mathbb{R}^2$. Which of the following subsets of $V$ is a subspace?
   1. (A) $\{(x, y) : x + y = 1\}$
   2. (B) $\{(x, y) : x \geq 0, y \geq 0\}$
   3. (C) $\{(x, y) : x = 0\}$
   4. (D) $\{(x, y) : x^2 + y^2 = 1\}$
3. Which of the following sets of vectors in $\mathbb{R}^3$ is linearly independent?
   1. (A) $\{(1, 0, 0), (0, 1, 0), (1, 1, 0)\}$
   2. (B) $\{(1, 2, 3), (2, 4, 6), (3, 6, 9)\}$
   3. (C) $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$
   4. (D) $\{(1, 1, 1), (2, 2, 2), (3, 3, 3)\}$
4. What is the dimension of the vector space of all $2 \times 2$ matrices?
   1. (A) 2
   2. (B) 3
   3. (C) 4
   4. (D) 5
5. Which of the following is a linear combination of the vectors $\mathbf{u} = (1, 2)$ and $\mathbf{v} = (3, 4)$?
   1. (A) (4, 6)
   2. (B) (0, 0)
   3. (C) (2, 3)
   4. (D) (5, 8)
6. If $V$ is a vector space, then for any scalar $c$ and vector $\mathbf{v} \in V$, $c\mathbf{0} = $?
   1. (A) $c$
   2. (B) $\mathbf{v}$
   3. (C) $\mathbf{0}$
   4. (D) $1$
7. Let $V = \mathbb{R}^3$. Does the set $\{(1, 0, 0), (0, 1, 0)\}$ span $V$?
   1. (A) Yes
   2. (B) No
   3. (C) Only if we include (0, 0, 1)
   4. (D) It depends on the basis