Math Problem Statement
Solution
The problem asks to find a nontrivial solution of the equation , where
The hint suggests thinking of this equation as a vector equation.
Approach:
To find a nontrivial solution, we need to determine a vector such that when multiplied by the matrix , it results in the zero vector, i.e.,
Since the matrix has dependent rows (as indicated by the fact that each row is a multiple of another), it will have infinitely many solutions, one of which is nontrivial. We can use inspection to find a nontrivial solution.
By inspection, let's check for a solution.It appears that my Python environment was reset, causing my previous variables to be lost. Let me redefine everything and find the nontrivial solution again.A nontrivial solution to the equation is .
This vector represents a direction in the null space of matrix . Any scalar multiple of this vector will also satisfy the equation , providing other nontrivial solutions.
Would you like more details or have any questions?
Here are some related questions you might find interesting:
- What does the null space of a matrix represent?
- How can you determine the number of solutions to ?
- How is the rank of matrix related to the dimension of its null space?
- What is the significance of finding a nontrivial solution to a homogeneous system of linear equations?
- How do row operations affect the null space of a matrix?
Tip: The null space gives all solutions to . For a matrix with dependent rows or columns, always expect nontrivial solutions.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Null Space
Formulas
Matrix-vector multiplication
Theorems
Rank-nullity theorem
Suitable Grade Level
Advanced Undergraduate