Math Problem Statement

4minus 3

Solution

To determine the value of xx for which the given vectors are not linearly independent, we must identify the condition under which the vectors become linearly dependent.

The vectors given in the image are: v1=[x3],v2=[1218].\mathbf{v_1} = \begin{bmatrix} x \\ 3 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} 12 \\ -18 \end{bmatrix}.

Two vectors v1\mathbf{v_1} and v2\mathbf{v_2} are linearly dependent if there exists a scalar λ\lambda such that: v1=λv2.\mathbf{v_1} = \lambda \mathbf{v_2}.

This means: [x3]=λ[1218].\begin{bmatrix} x \\ 3 \end{bmatrix} = \lambda \begin{bmatrix} 12 \\ -18 \end{bmatrix}.

Expanding this equation, we get two separate equations: x=12λ,3=18λ.x = 12\lambda, \quad 3 = -18\lambda.

From the second equation: λ=318=16.\lambda = \frac{3}{-18} = -\frac{1}{6}.

Substitute λ=16\lambda = -\frac{1}{6} into the first equation: x=12(16)=2.x = 12 \left(-\frac{1}{6}\right) = -2.

Thus, the vectors are not linearly independent (i.e., they are linearly dependent) when x=2x = -2.

Answer: 2-2.

Would you like more details on this solution or have any questions?

Related Questions:

  1. How can we determine if three vectors are linearly independent or dependent?
  2. What is the geometric interpretation of linearly dependent vectors?
  3. How do we find the determinant of a matrix formed by vectors to check independence?
  4. What does it mean for vectors to span a vector space?
  5. How does the concept of linear independence extend to higher dimensions?

Tip: Remember, two vectors are linearly dependent if they lie on the same line through the origin.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vectors
Linear Dependence

Formulas

Vector equality in terms of scalar multiplication

Theorems

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Suitable Grade Level

Advanced High School