Solved Examples: Finding a Basis for the Null Space and Column Space

📚 Quick Study Guide

• 🔢 Null Space: The null space of a matrix $A$ is the set of all vectors $x$ such that $Ax = 0$. To find a basis, solve $Ax = 0$ and express the solution in parametric vector form. The vectors in the parametric form constitute the basis.
• 📈 Column Space: The column space of a matrix $A$ is the span of the columns of $A$. To find a basis, reduce $A$ to echelon form and identify the pivot columns. The corresponding columns in the original matrix $A$ form the basis for the column space.
• 💡 Key Idea: The null space is a subspace of $\mathbb{R}^n$ where $n$ is the number of columns of $A$, and the column space is a subspace of $\mathbb{R}^m$ where $m$ is the number of rows of $A$.
• 📝 Parametric Vector Form: Express solutions to $Ax=0$ in terms of free variables. For example, if $x_3$ is free, you'll have $x = x_3 \cdot v$ for some vector $v$. This $v$ is part of your null space basis.
• 🔍 Pivot Columns: These are the columns in the reduced row echelon form (RREF) that contain a leading 1. They tell you which columns in the original matrix form the basis for the column space.

Practice Quiz

1. Question 1: What is the null space of a matrix $A$?
   1. A) The set of all vectors $b$ such that $Ax = b$ has a solution.
   2. B) The set of all vectors $x$ such that $Ax = 0$.
   3. C) The span of the rows of $A$.
   4. D) The set containing only the zero vector.
2. Question 2: How do you find a basis for the null space of a matrix $A$?
   1. A) Find the pivot columns of $A$.
   2. B) Solve $Ax = b$ for some vector $b$.
   3. C) Solve $Ax = 0$ and express the solution in parametric vector form.
   4. D) Find the eigenvalues of $A$.
3. Question 3: What is the column space of a matrix $A$?
   1. A) The set of all vectors $x$ such that $Ax = 0$.
   2. B) The span of the rows of $A$.
   3. C) The span of the columns of $A$.
   4. D) The set containing only the zero vector.
4. Question 4: How do you find a basis for the column space of a matrix $A$?
   1. A) Solve $Ax = 0$.
   2. B) Reduce $A$ to echelon form and identify the pivot columns; the corresponding columns in the original matrix $A$ form the basis.
   3. C) Find the eigenvalues of $A$.
   4. D) Find the left null space of $A$.
5. Question 5: If $A$ is an $m \times n$ matrix, the null space of $A$ is a subspace of which vector space?
   1. A) $\mathbb{R}^m$
   2. B) $\mathbb{R}^n$
   3. C) $\mathbb{R}^{m \times n}$
   4. D) $\mathbb{R}$
6. Question 6: If $A$ is an $m \times n$ matrix, the column space of $A$ is a subspace of which vector space?
   1. A) $\mathbb{R}^n$
   2. B) $\mathbb{R}^m$
   3. C) $\mathbb{R}^{m \times n}$
   4. D) $\mathbb{R}$
7. Question 7: What are pivot columns useful for when finding bases?
   1. A) Finding a basis for the null space.
   2. B) Finding a basis for the row space.
   3. C) Finding a basis for the column space.
   4. D) Finding the eigenvalues.