https://file.aiquickdraw.com/mathbot/file/1725927296303t359ggre.jpg

We are given a linear transformation \( L : \mathbb{R}^3 \to \mathbb{R}^3 \) represented by the matrix

\[
A = \begin{pmatrix}
0 & 3 & 1 \\
2 & -1 & 4 \\
2 & -7 & 2
\end{pmatrix}.
\]

To find the kernel (null space) of \( L \), we must solve for the set of vectors \( \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \in \mathbb{R}^3 \) such that:

\[
A\mathbf{x} = \mathbf{0}.
\]

This gives us the homogeneous system of equations:

\[
\begin{pmatrix}
0 & 3 & 1 \\
2 & -1 & 4 \\
2 & -7 & 2
\end{pmatrix}
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.
\]

This leads to the following system of equations:

1. \( 0x_1 + 3x_2 + x_3 = 0 \), or \( 3x_2 + x_3 = 0 \)
2. \( 2x_1 - x_2 + 4x_3 = 0 \)
3. \( 2x_1 - 7x_2 + 2x_3 = 0 \)

### Step 1: Solve the first equation

From the first equation \( 3x_2 + x_3 = 0 \), we can express \( x_3 \) in terms of \( x_2 \):

\[
x_3 = -3x_2.
\]

### Step 2: Substitute \( x_3 = -3x_2 \) into the remaining equations

Substitute this into the second and third equations:

- Second equation: \( 2x_1 - x_2 + 4(-3x_2) = 0 \)
\[
2x_1 - x_2 - 12x_2 = 0 \quad \Rightarrow \quad 2x_1 - 13x_2 = 0 \quad \Rightarrow \quad x_1 = \frac{13}{2}x_2.
\]

- Third equation: \( 2x_1 - 7x_2 + 2(-3x_2) = 0 \)
\[
2x_1 - 7x_2 - 6x_2 = 0 \quad \Rightarrow \quad 2x_1 - 13x_2 = 0.
\]
This is the same equation as the second, so no new information is obtained.

### Step 3: General solution

The general solution is:

\[
\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} \frac{13}{2}x_2 \\ x_2 \\ -3x_2 \end{pmatrix} = x_2 \begin{pmatrix} \frac{13}{2} \\ 1 \\ -3 \end{pmatrix}.
\]

Thus, the kernel (null space) of \( L \) is spanned by the vector:

\[
\ker(L) = \left\{ \lambda \begin{pmatrix} \frac{13}{2} \\ 1 \\ -3 \end{pmatrix} : \lambda \in \mathbb{C} \right\}.
\]

Would you like more details or have any other questions?

### Related Questions:
1. What is the rank of the matrix \( A \)?
2. How can we find the image (column space) of the linear transformation \( L \)?
3. How do we determine whether \( L \) is injective or surjective?
4. How do eigenvalues and eigenvectors relate to the kernel of a matrix?
5. How can we compute the determinant of \( A \) and what does it tell us about \( L \)?

### Tip:
Always check the consistency of your system of equations when solving for the kernel, especially when multiple equations reduce to the same form.

Find the kernel (null space) of the linear transformation given by the matrix A = [[0, 3, 1], [2, -1, 4], [2, -7, 2]].

Learn how to find the kernel (null space) of a linear transformation defined by the matrix A = [[0, 3, 1], [2, -1, 4], [2, -7, 2]]. This problem covers key concepts in linear algebra, including solving homogeneous systems and determining the kernel. Suitable for college-level linear algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation