Math Problem Statement
Show that the vectors (2, 3, 1)T , (2, 1, 3)T and (1, 1, 1)T are linearly dependent in R3
Solution
To show that the vectors , , and are linearly dependent, we need to prove that there exist scalars , , and , not all zero, such that:
This is equivalent to solving the equation:
Writing this out in component form gives the following system of equations:
2c_1 + 2c_2 + c_3 = 0 \tag{1} 3c_1 + c_2 + c_3 = 0 \tag{2} c_1 + 3c_2 + c_3 = 0 \tag{3}
Step 1: Solve the system of equations
We can solve this system using standard algebraic techniques, such as substitution or elimination. Let's use elimination here.
Subtract equation (1) from equation (2): c_1 - c_2 = 0 \quad \Rightarrow \quad c_1 = c_2 \tag{4}
Now subtract equation (1) from equation (3): -c_1 + c_2 = 0 \quad \Rightarrow \quad c_1 = c_2 \tag{5}
Step 2: Substitute into one of the original equations
Substitute into equation (1):
Step 3: Conclusion
We have and . Therefore, the solution to the system is:
where is any non-zero scalar. Since not all the coefficients are zero, the vectors are linearly dependent.
Would you like more details or have any further questions? Here are some related topics:
- What does it mean for vectors to be linearly independent?
- How can we use matrix methods to check for linear dependence?
- What is the geometric interpretation of linearly dependent vectors?
- How does the rank of a matrix relate to linear dependence?
- Can we extend this concept to higher dimensions?
Tip: The easiest way to check for linear dependence in is to see if the determinant of the matrix formed by the vectors is zero!
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Spaces
Linear Dependence
Formulas
c1 * v1 + c2 * v2 + c3 * v3 = 0
System of equations: 2c1 + 2c2 + c3 = 0, 3c1 + c2 + c3 = 0, c1 + 3c2 + c3 = 0
Theorems
Linear Dependence Theorem
System of Equations
Suitable Grade Level
Undergraduate level (Linear Algebra)
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