Solved Examples: Finding a Basis for R3

📚 Quick Study Guide

• 📐 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.
• ➕ 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 $c_1 = c_2 = ... = c_n = 0$.
• 🔑 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 {$v_1, v_2, ..., v_n$}.
• 📏 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!
• 💡 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.