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Solved Examples: Finding a Basis for R3

Hey there! ๐Ÿ‘‹ Finding a basis for R3 can seem tricky, but it's actually quite straightforward once you understand the key concepts. This study guide and quiz will help you nail it! Let's dive in! ๐Ÿงฎ
๐Ÿงฎ Mathematics

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wagner.claire99 Dec 27, 2025

๐Ÿ“š Quick Study Guide

  • ๐Ÿ“ Definition of Basis: A basis for $\mathbb{R}^3$ is a set of vectors that are linearly independent and span $\mathbb{R}^3$. This means any vector in $\mathbb{R}^3$ can be written as a linear combination of the basis vectors.
  • โž• Linear Independence: A set of vectors {$v_1, v_2, ..., v_n$} is linearly independent if the only solution to the equation $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ is $c_1 = c_2 = ... = c_n = 0$.
  • ๐Ÿ”‘ Spanning Set: A set of vectors {$v_1, v_2, ..., v_n$} spans $\mathbb{R}^3$ if every vector in $\mathbb{R}^3$ can be written as a linear combination of {$v_1, v_2, ..., v_n$}.
  • ๐Ÿ“ Standard Basis: The standard basis for $\mathbb{R}^3$ is {$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$}. It's a very common and useful basis!
  • ๐Ÿ’ก Dimension: The dimension of $\mathbb{R}^3$ is 3, meaning any basis for $\mathbb{R}^3$ must contain exactly 3 vectors.

Practice Quiz

  1. Question 1: Which of the following sets of vectors forms a basis for $\mathbb{R}^3$?
    1. A) {$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$}
    2. B) {$\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, $\begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix}$, $\begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix}$}
    3. C) {$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$}
    4. D) {$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$, $\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$}
  2. Question 2: Which condition is necessary for a set of vectors to be a basis for $\mathbb{R}^3$?
    1. A) The vectors must be orthogonal.
    2. B) The vectors must be linearly dependent.
    3. C) The vectors must span $\mathbb{R}^3$.
    4. D) The vectors must have a magnitude of 1.
  3. Question 3: Determine if the following set of vectors is linearly independent: {$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$, $\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$}.
    1. A) Linearly Dependent
    2. B) Linearly Independent
    3. C) Cannot be determined
    4. D) Orthogonal
  4. Question 4: How many vectors are required to form a basis for $\mathbb{R}^3$?
    1. A) 2
    2. B) 3
    3. C) 4
    4. D) Any number of vectors.
  5. Question 5: Which of the following sets does NOT span $\mathbb{R}^3$?
    1. A) {$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$}
    2. B) {$\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, $\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$}
    3. C) {$\begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 3 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 0 \\ 5 \end{bmatrix}$}
    4. D) {$\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, $\begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix}$, $\begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix}$}
  6. Question 6: Which of the following statements is correct regarding a basis for $\mathbb{R}^3$?
    1. A) It must contain the zero vector.
    2. B) It must be unique.
    3. C) It must be linearly independent and span $\mathbb{R}^3$.
    4. D) It must contain orthogonal vectors.
  7. Question 7: Determine if the following set of vectors forms a basis for $\mathbb{R}^3$: {$\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$, $\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$}.
    1. A) Yes, it forms a basis.
    2. B) No, it does not form a basis.
    3. C) Cannot be determined with the given information.
    4. D) The set is orthogonal.
Click to see Answers
  1. C
  2. C
  3. B
  4. B
  5. D
  6. C
  7. A

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