Math Problem Statement

Are the vectors [3 1 -4], [4 -1 -3], and [-5 5 1] linearly independent? If they are linearly dependent, find scalars that are not all zero such that the equation below is true. If they are linearly independent, enter all zeros in the equation below.

Solution

The problem in the image asks if the vectors

v1=[31−4],v2=[4−1−3],v3=[−551]\mathbf{v}_1 = \begin{bmatrix} 3 \\ 1 \\ -4 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 4 \\ -1 \\ -3 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} -5 \\ 5 \\ 1 \end{bmatrix}

are linearly independent or dependent.

To determine this, we need to check if the only solution to the equation

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

is c1=c2=c3=0c_1 = c_2 = c_3 = 0 (which would imply linear independence), or if there are non-zero scalars c1,c2,c3c_1, c_2, c_3 that satisfy the equation (which would imply linear dependence).

This system can be represented as a matrix equation:

3 & 4 & -5 \\ 1 & -1 & 5 \\ -4 & -3 & 1 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$ We will perform Gaussian elimination or compute the determinant to check if the vectors are linearly independent. Let me compute this.The determinant of the matrix formed by the vectors is approximately $$-7$$. Since the determinant is non-zero, this means the vectors are **linearly independent**. If you have any further questions, or would like a detailed explanation of Gaussian elimination or the properties of determinants, feel free to ask! Here are five related questions to deepen understanding: 1. How does the determinant help in determining linear independence? 2. What happens when the determinant of a matrix is zero? 3. Can you explain how Gaussian elimination works for this problem? 4. What are the geometric interpretations of linearly independent vectors? 5. What is the relationship between linear dependence and the rank of a matrix? **Tip**: In general, three vectors in $$\mathbb{R}^3$$ are linearly independent if they form the basis of the 3D space, meaning no vector lies in the plane defined by the other two.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Linear Independence
Matrix Operations

Formulas

Matrix Equation: c1*v1 + c2*v2 + c3*v3 = 0
Determinant Calculation

Theorems

Linear Independence Theorem
Gaussian Elimination

Suitable Grade Level

Grades 11-12 or College Level

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