Math Problem Statement
Find the kernel (null space) of the linear transformation given by the matrix A = [[0, 3, 1], [2, -1, 4], [2, -7, 2]].
Solution
We are given a linear transformation represented by the matrix
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Kernel
Null Space
Systems of Linear Equations
Formulas
Ax = 0 (homogeneous system)
x_3 = -3x_2
x_1 = (13/2)x_2
Theorems
Kernel of a linear transformation
Solving homogeneous systems of linear equations
Suitable Grade Level
College-Level Linear Algebra