Math Problem Statement

Determine whether the vectors v_1 = [1, 2, 3], v_2 = [3, 1, 5], v_3 = [2, 5, 1] are linearly independent. Justify your answer.

Solution

To determine whether the vectors v1=[123],v2=[315],v3=[251]v_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, v_2 = \begin{bmatrix} 3 \\ 1 \\ 5 \end{bmatrix}, v_3 = \begin{bmatrix} 2 \\ 5 \\ 1 \end{bmatrix} are linearly independent, we need to set up and solve the following equation:

c1v1+c2v2+c3v3=0c_1 v_1 + c_2 v_2 + c_3 v_3 = 0

This can be written as the system of equations:

c1[123]+c2[315]+c3[251]=[000]c_1 \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + c_2 \begin{bmatrix} 3 \\ 1 \\ 5 \end{bmatrix} + c_3 \begin{bmatrix} 2 \\ 5 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}

This gives us the following system of linear equations:

1. & \quad c_1 + 3c_2 + 2c_3 = 0 \\ 2. & \quad 2c_1 + c_2 + 5c_3 = 0 \\ 3. & \quad 3c_1 + 5c_2 + c_3 = 0 \end{aligned}$$ We can solve this system using Gaussian elimination or by calculating the determinant of the matrix formed by $$v_1$$, $$v_2$$, and $$v_3$$. If the determinant is non-zero, the vectors are linearly independent. If the determinant is zero, they are linearly dependent. The matrix formed by the vectors is: $$A = \begin{bmatrix} 1 & 3 & 2 \\ 2 & 1 & 5 \\ 3 & 5 & 1 \end{bmatrix}$$ Now, let's compute the determinant of matrix $$A$$. $$\det(A) = \begin{vmatrix} 1 & 3 & 2 \\ 2 & 1 & 5 \\ 3 & 5 & 1 \end{vmatrix}$$ I'll calculate the determinant next.The determinant of the matrix $$A$$ is 29, which is non-zero. Therefore, the vectors $$v_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, v_2 = \begin{bmatrix} 3 \\ 1 \\ 5 \end{bmatrix}, v_3 = \begin{bmatrix} 2 \\ 5 \\ 1 \end{bmatrix}$$ are **linearly independent**. Would you like further details or have any questions? Here are 5 related questions to expand on this topic: 1. What does it mean geometrically for three vectors to be linearly independent? 2. How would the solution change if the determinant had been zero? 3. What are some methods other than determinants to check for linear independence? 4. Can you explain the significance of linear independence in the context of vector spaces? 5. How would this method apply if you had four or more vectors? **Tip:** When checking for linear independence of multiple vectors, the number of vectors should be equal to or less than the dimension of the space they inhabit for them to potentially be independent.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Linear Independence
Determinants

Formulas

c_1 * v_1 + c_2 * v_2 + c_3 * v_3 = 0
Determinant of a 3x3 matrix formula: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

Theorems

Linear Independence Theorem
Determinant Theorem

Suitable Grade Level

Undergraduate or Advanced High School (Grades 11-12)

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