Math Problem Statement

briefly explain the problem

Write \(\mathbf{v} = (2, -5, 3)\) as a linear combination of the vectors \(\mathbf{a}_1 = (1, -3, 2)\), \(\mathbf{a}_2 = (2, -4, -1)\), and \(\mathbf{a}_3 = (1, -5, 7)\).

Solution

The problem in the image is about expressing the vector v=(2,−5,3)\mathbf{v} = (2, -5, 3) as a linear combination of the given vectors a1=(1,−3,2)\mathbf{a}_1 = (1, -3, 2), a2=(2,−4,−1)\mathbf{a}_2 = (2, -4, -1), and a3=(1,−5,7)\mathbf{a}_3 = (1, -5, 7).

To find if v\mathbf{v} can be expressed as a linear combination of these vectors, we need to solve the system of equations that corresponds to:

xa1+ya2+za3=vx\mathbf{a}_1 + y\mathbf{a}_2 + z\mathbf{a}_3 = \mathbf{v}

This results in the following system of linear equations:

x + 2y + z &= 2, \\ -3x - 4y - 5z &= -5, \\ 2x - y + 7z &= 3. \end{align*}$$ The solution involves comparing corresponding components and forming these equations. The calculations reveal that the third equation does not align with the other two, leading to an inconsistency ($$0 = 1$$). This inconsistency means that the system has no solution, and thus $$\mathbf{v}$$ cannot be written as a linear combination of $$\mathbf{a}_1$$, $$\mathbf{a}_2$$, and $$\mathbf{a}_3$$. Would you like further details on any part of the solution? Here are some related questions: 1. What is a linear combination in vector spaces? 2. How do you determine if a vector is a linear combination of other vectors? 3. What does it mean when a system of equations has no solution? 4. How can you represent a system of equations using matrices? 5. What are the implications of an inconsistent system in linear algebra? **Tip:** Always check the determinant or use row reduction methods to identify inconsistencies in a system of linear equations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Linear Combinations
Systems of Linear Equations

Formulas

x\mathbf{a}_1 + y\mathbf{a}_2 + z\mathbf{a}_3 = \mathbf{v}
Solving Systems of Linear Equations

Theorems

Consistency of a System of Linear Equations

Suitable Grade Level

Grades 11-12

Related Recommendation

Linearly Dependent Vectors {v1, v2, v3} and Non-trivial Linear Combination

Determine if Vectors Lie in the Span of Given Vectors using Linear Combinations

Express Vector (-4, 7, 1) as a Linear Combination of Given Vectors in R^3

Determine if Vector b is a Linear Combination of Vectors a1 and a2

Linear Combination of Vectors: Expressing w as c₁ * v₁ + c₂ * v₂