Math Problem Statement

Determine if b is a linear combination of v₁, v₂, and v₃, where v₁ = [2, 1, 5], v₂ = [1, 2, 14], v₃ = [-11, -10, -62], and b = [19, 17, 103].

Solution

We are tasked with determining if vector b is a linear combination of the vectors v₁, v₂, and v₃. This can be done by solving the matrix equation:

c1v1+c2v2+c3v3=bc_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3} = \mathbf{b}

In matrix form, this can be written as:

\mathbf{v_1} & \mathbf{v_2} & \mathbf{v_3} \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \mathbf{b}$$ Substituting the vectors: $$\begin{bmatrix} 2 & 1 & -11 \\ 1 & 2 & -10 \\ 5 & 14 & -62 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 19 \\ 17 \\ 103 \end{bmatrix}$$ This gives a system of equations: $$2c_1 + 1c_2 - 11c_3 = 19$$ $$1c_1 + 2c_2 - 10c_3 = 17$$ $$5c_1 + 14c_2 - 62c_3 = 103$$ I will now solve this system to determine if there are values for $$c_1$$, $$c_2$$, and $$c_3$$ that satisfy these equations.The system of equations does not have a solution, meaning that vector **b** is **not** a linear combination of the vectors **v₁**, **v₂**, and **v₃**. Would you like further details on the process or have any specific questions? Here are five related questions that might interest you: 1. What does it mean for a vector to be a linear combination of other vectors? 2. How can we check if a set of vectors is linearly independent? 3. What is the geometric interpretation of linear combinations? 4. How do we solve systems of linear equations using matrices? 5. What is the importance of the rank of a matrix in solving linear equations? **Tip:** If the system of equations has no solution, it suggests that the vector does not lie in the span of the other vectors, which means the vectors do not cover all the space required to represent it.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Linear Combinations
Matrix Equations

Formulas

c₁v₁ + c₂v₂ + c₃v₃ = b
Matrix equation: A * c = b, where A = [v₁ v₂ v₃] and c = [c₁, c₂, c₃]

Theorems

System of Linear Equations
Matrix Rank Theorem

Suitable Grade Level

Undergraduate

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