https://file.aiquickdraw.com/mathbot/file/17283421719753760r6bo.jpg

The problem asks you to express the vector **w** as a linear combination of the vectors **v₁** and **v₂**.

Given:
\[
\mathbf{w} = \begin{bmatrix} -8 \\ 11 \\ 8 \end{bmatrix}, \quad \mathbf{v_1} = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}
\]

We need to find scalars \(c_1\) and \(c_2\) such that:
\[
\mathbf{w} = c_1 \mathbf{v_1} + c_2 \mathbf{v_2}
\]
This can be written as the system of equations:
\[
\begin{bmatrix} -8 \\ 11 \\ 8 \end{bmatrix} = c_1 \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}
\]
This gives the following system of equations:
1. \(-c_1 + 2c_2 = -8\)
2. \(c_1 + c_2 = 11\)
3. \(c_1 + c_2 = 8\)

However, notice that equations 2 and 3 are contradictory, because both cannot be true simultaneously. This means there is no solution for \(c_1\) and \(c_2\) that satisfies all three equations.

Therefore, the vector **w** cannot be expressed as a linear combination of **v₁** and **v₂**, and the answer is **DNE** (Does Not Exist).

Would you like further details or have any questions about this? Here are 5 related questions for further exploration:

1. How do you solve a system of equations for linear combinations?
2. What does it mean if a vector cannot be expressed as a linear combination of other vectors?
3. What is the geometric interpretation of a linear combination in 3D space?
4. How would you check if a set of vectors is linearly independent?
5. What is the rank of a matrix, and how is it related to linear combinations?

**Tip**: When equations in a system contradict each other, it typically means the vectors are linearly independent and the combination does not exist.

Express the vector w as a linear combination of the vectors v₁ and v₂, or determine if it cannot be expressed as such.

Learn how to determine if a vector w can be expressed as a linear combination of two vectors v₁ and v₂. This problem involves solving a system of linear equations and covers core concepts in Linear Algebra, suitable for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation