### Linear Combinations and Vector Spaces**Learning Objective:** Understand how to express a vector as a linear combination of a set of vectors.---#### Problem StatementThe vectors $ \mathbf{v}_1 = \begin{bmatrix} 1 \\ -4 \end{bmatrix} $, $ \mathbf{v}_2 = \begin{bmatrix} 2 \\ -11 \end{bmatrix} $, and $ \mathbf{v}_3 = \begin{bmatrix} 0 \\ -3 \end{bmatrix} $ span $\mathbb{R}^2$ but do not form a basis. **Task:** Find two different ways to express $\begin{bmatrix} -3 \\ 24 \end{bmatrix}$ as a linear combination of $ \mathbf{v}_1 $, $ \mathbf{v}_2 $, and $ \mathbf{v}_3 $.#### ApproachWrite $\begin{bmatrix} -3 \\ 24 \end{bmatrix}$ as a linear combination of $ \mathbf{v}_1 $, $ \mathbf{v}_2 $, and $ \mathbf{v}_3 $ when the coefficient of $ \mathbf{v}_3 $ is 0.\[\begin{bmatrix} -3 \\ 24 \end{bmatrix} = \_ \cdot \mathbf{v}_1 + \_ \cdot \mathbf{v}_2\]This leads to the following equation:\[\begin{bmatrix} -3 \\ 24 \end{bmatrix} = c_1 \begin{bmatrix} 1 \\ -4 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ -11 \end{bmatrix}\]To solve for $c_1$ and $c_2$, set up the following system of linear equations:\[-3 = c_1 + 2c_2\]\[24 = -4c_1 - 11c_2\]Solving the above system will provide the coefficients $c_1$ and $c_2$.**Note:** There might be multiple correct solutions due to the vectors spanning $\mathbb{R}^2$ but not forming a basis.

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The vectors v.= span R but do not form a basis. Find two different ways to express -3 - 3 as a linear combination of V V V 24 v = Write as a linear combination of v. v. V when the coefficient of V is 0. 24 [2అ-ం·

The vectors v.= span R but do not form a basis. Find two different ways to express -3 - 3 as a linear combination of V V V 24 v = Write as a linear combination of v. v. V when the coefficient of V is 0. 24 [2అ-ం·

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

### Linear Combinations and Vector Spaces

**Learning Objective:** Understand how to express a vector as a linear combination of a set of vectors.

---
#### Problem Statement

The vectors $ \mathbf{v}_1 = \begin{bmatrix} 1 \\ -4 \end{bmatrix} $, $ \mathbf{v}_2 = \begin{bmatrix} 2 \\ -11 \end{bmatrix} $, and $ \mathbf{v}_3 = \begin{bmatrix} 0 \\ -3 \end{bmatrix} $ span $\mathbb{R}^2$ but do not form a basis.

**Task:** Find two different ways to express $\begin{bmatrix} -3 \\ 24 \end{bmatrix}$ as a linear combination of $ \mathbf{v}_1 $, $ \mathbf{v}_2 $, and $ \mathbf{v}_3 $.

#### Approach

Write $\begin{bmatrix} -3 \\ 24 \end{bmatrix}$ as a linear combination of $ \mathbf{v}_1 $, $ \mathbf{v}_2 $, and $ \mathbf{v}_3 $ when the coefficient of $ \mathbf{v}_3 $ is 0.

\[
\begin{bmatrix} -3 \\ 24 \end{bmatrix} = \_ \cdot \mathbf{v}_1 + \_ \cdot \mathbf{v}_2
\]

This leads to the following equation:

\[
\begin{bmatrix} -3 \\ 24 \end{bmatrix} = c_1 \begin{bmatrix} 1 \\ -4 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ -11 \end{bmatrix}
\]

To solve for $c_1$ and $c_2$, set up the following system of linear equations:

\[
-3 = c_1 + 2c_2
\]
\[
24 = -4c_1 - 11c_2
\]

Solving the above system will provide the coefficients $c_1$ and $c_2$.

**Note:** There might be multiple correct solutions due to the vectors spanning $\mathbb{R}^2$ but not forming a basis.

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