stephanieanthony1985
stephanieanthony1985 5d ago โ€ข 0 views

Test Your Knowledge: Basis for a Vector Space Exam Questions

Hey there! ๐Ÿ‘‹ Getting ready for your vector space exam? No sweat! I've put together a quick study guide and a practice quiz to help you ace it. Let's dive in and make sure you're totally prepped! ๐Ÿง 
๐Ÿงฎ Mathematics

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jessica.white Dec 27, 2025

๐Ÿ“š Quick Study Guide

  • โž• Vector Space Definition: A set $V$ with two operations (addition and scalar multiplication) satisfying certain axioms.
  • ๐Ÿ”ข 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 $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$
  • โž— 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 $\mathbf{u} \in V$
  • ๐Ÿ“ Subspace: A subset $W$ of $V$ is a subspace if it is closed under vector addition and scalar multiplication.
  • ๐Ÿ’ก 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 $c_1, c_2, ..., c_n$.
  • ๐Ÿ”‘ Span: The span of a set of vectors is the set of all linear combinations of those vectors.
  • ๐Ÿ“ˆ 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 $c_1 = c_2 = ... = c_n = 0$.
  • ๐ŸŽฏ Basis: A basis for a vector space $V$ is a set of linearly independent vectors that span $V$.
  • ๐Ÿ“ 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
Click to see Answers
  1. B
  2. C
  3. C
  4. C
  5. D
  6. C
  7. B

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