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 v1=(231)\mathbf{v}_1 = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}, v2=(213)\mathbf{v}_2 = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}, and v3=(111)\mathbf{v}_3 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} are linearly dependent, we need to prove that there exist scalars c1c_1, c2c_2, and c3c_3, not all zero, such that:

c1v1+c2v2+c3v3=0c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3 \mathbf{v}_3 = \mathbf{0}

This is equivalent to solving the equation:

c1(231)+c2(213)+c3(111)=(000)c_1 \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} + c_2 \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix} + c_3 \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}

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): (3c1+c2+c3)(2c1+2c2+c3)=0(3c_1 + c_2 + c_3) - (2c_1 + 2c_2 + c_3) = 0 c_1 - c_2 = 0 \quad \Rightarrow \quad c_1 = c_2 \tag{4}

Now subtract equation (1) from equation (3): (c1+3c2+c3)(2c1+2c2+c3)=0(c_1 + 3c_2 + c_3) - (2c_1 + 2c_2 + c_3) = 0 -c_1 + c_2 = 0 \quad \Rightarrow \quad c_1 = c_2 \tag{5}

Step 2: Substitute c1=c2c_1 = c_2 into one of the original equations

Substitute c1=c2c_1 = c_2 into equation (1): 2c1+2c1+c3=02c_1 + 2c_1 + c_3 = 0 4c1+c3=0c3=4c14c_1 + c_3 = 0 \quad \Rightarrow \quad c_3 = -4c_1

Step 3: Conclusion

We have c1=c2c_1 = c_2 and c3=4c1c_3 = -4c_1. Therefore, the solution to the system is:

c1=t,c2=t,c3=4tc_1 = t, \quad c_2 = t, \quad c_3 = -4t

where tt is any non-zero scalar. Since not all the coefficients c1,c2,c3c_1, c_2, c_3 are zero, the vectors are linearly dependent.


Would you like more details or have any further questions? Here are some related topics:

  1. What does it mean for vectors to be linearly independent?
  2. How can we use matrix methods to check for linear dependence?
  3. What is the geometric interpretation of linearly dependent vectors?
  4. How does the rank of a matrix relate to linear dependence?
  5. Can we extend this concept to higher dimensions?

Tip: The easiest way to check for linear dependence in R3\mathbb{R}^3 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)