Math Problem Statement

Write [4 -3 -2] as a linear combination of the vectors [1 -1 0], [-3 2 5], [3 4 -1].

Solution

The problem is asking to express the vector [432]\begin{bmatrix} 4 \\ -3 \\ -2 \end{bmatrix} as a linear combination of three given vectors:

[110],[325],[341]\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} -3 \\ 2 \\ 5 \end{bmatrix}, \quad \begin{bmatrix} 3 \\ 4 \\ -1 \end{bmatrix}

We need to find scalars c1c_1, c2c_2, and c3c_3 such that:

[432]=c1[110]+c2[325]+c3[341]\begin{bmatrix} 4 \\ -3 \\ -2 \end{bmatrix} = c_1 \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} + c_2 \begin{bmatrix} -3 \\ 2 \\ 5 \end{bmatrix} + c_3 \begin{bmatrix} 3 \\ 4 \\ -1 \end{bmatrix}

This leads to the following system of equations by equating the components of the vectors:

  1. 1c13c2+3c3=41c_1 - 3c_2 + 3c_3 = 4
  2. 1c1+2c2+4c3=3-1c_1 + 2c_2 + 4c_3 = -3
  3. 0c1+5c21c3=20c_1 + 5c_2 - 1c_3 = -2

I will solve this system to find c1c_1, c2c_2, and c3c_3.The solution to the system is:

c1=4417,c2=1334,c3=334c_1 = \frac{44}{17}, \quad c_2 = -\frac{13}{34}, \quad c_3 = \frac{3}{34}

Thus, the vector [432]\begin{bmatrix} 4 \\ -3 \\ -2 \end{bmatrix} can be written as:

4417[110]1334[325]+334[341]\frac{44}{17} \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} - \frac{13}{34} \begin{bmatrix} -3 \\ 2 \\ 5 \end{bmatrix} + \frac{3}{34} \begin{bmatrix} 3 \\ 4 \\ -1 \end{bmatrix}

Let me know if you'd like more details on the solution process!

Related questions:

  1. How do you check if a set of vectors is linearly independent?
  2. What does it mean for vectors to form a basis?
  3. Can any vector be written as a linear combination of other vectors?
  4. How is the rank of a matrix related to linear combinations?
  5. How does the concept of linear combinations relate to vector spaces?

Tip: When solving systems of equations, arranging them in matrix form can make the process clearer and help visualize the solution!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Linear Combinations
Systems of Equations

Formulas

Linear combination: c1 * v1 + c2 * v2 + c3 * v3 = result_vector
System of equations derived from vector components

Theorems

Solving systems of linear equations
Matrix representation of linear systems

Suitable Grade Level

College level or advanced high school (Grades 11-12)