Math Problem Statement
Let A
⎡ ⎢ ⎣ 1 2 − 6 2 0 0 3 − 12 0 0 0 0 ⎤ ⎥ ⎦ . What is the solution space of A ?
Solution
The given matrix is:
1 & 2 & -6 \\ 2 & 0 & 0 \\ 3 & -12 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ We are tasked with determining the solution space of this matrix, which involves finding the null space of $$A$$, i.e., the set of vectors $$\mathbf{x}$$ that satisfy: $$A \mathbf{x} = 0$$ ### Step 1: Write the system of equations Let's assume $$\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$$. We can write the matrix equation $$A \mathbf{x} = 0$$ as the following system of equations: $$\begin{aligned} 1x_1 + 2x_2 - 6x_3 &= 0 \\ 2x_1 &= 0 \\ 3x_1 - 12x_2 &= 0 \end{aligned}$$ ### Step 2: Solve the system - From the second equation: $$2x_1 = 0$$, we have $$x_1 = 0$$. - Substituting $$x_1 = 0$$ into the first equation: $$0 + 2x_2 - 6x_3 = 0$$, which simplifies to: $$2x_2 - 6x_3 = 0 \implies x_2 = 3x_3$$ - The third equation becomes $$3(0) - 12x_2 = 0$$, which is automatically satisfied. ### Step 3: Solution space Thus, the solution space (null space) is given by: $$\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 3x_3 \\ x_3 \end{pmatrix} = x_3 \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}$$ The null space is spanned by the vector $$\begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}$$. ### Conclusion The solution space of the matrix $$A$$ is the set of all scalar multiples of the vector $$\begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}$$. In other words, the null space of $$A$$ is: $$\text{Null}(A) = \text{span} \left( \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix} \right)$$ Would you like more details or have any questions? ### Related questions: 1. What is the rank of matrix $$A$$? 2. How does one determine the nullity of a matrix? 3. Can a non-square matrix have a null space? If so, how is it found? 4. What are the implications of a matrix having a non-trivial null space? 5. How does the null space relate to the kernel of a linear transformation? ### Tip: The dimension of the null space of a matrix is called the **nullity**, and it provides insight into the "degree of freedom" in the system of linear equations represented by the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Null Space
Matrix Equations
Vector Spaces
Formulas
A \mathbf{x} = 0
Solution of linear systems: x_1 + 2x_2 - 6x_3 = 0, 2x_1 = 0, 3x_1 - 12x_2 = 0
Null Space = span(\begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix})
Theorems
Null Space Theorem
Rank-Nullity Theorem
Suitable Grade Level
College Level