Math Problem Statement

Let \(\vec{x} = \begin{bmatrix} 2 \\ 5 \\ 8 \\ 3 \end{bmatrix}\), \(\vec{y} = \begin{bmatrix} 2 \\ -1 \\ 0 \\ 5 \end{bmatrix}\), and \(\vec{z} = \begin{bmatrix} -1 \\ 2 \\ 2 \\ -3 \end{bmatrix}\) be vectors in \(\mathbb{R}^4\). Show that \(\vec{x}\) is in the span of \(\{\vec{y}, \vec{z}\}\) by writing it as a linear combination of \(\vec{y}\) and \(\vec{z}\).

Solution

To show that x⃗\vec{x} is in the span of {y⃗,z⃗}\{\vec{y}, \vec{z}\}, we need to express x⃗\vec{x} as a linear combination of y⃗\vec{y} and z⃗\vec{z}. In other words, we are looking for scalars aa and bb such that:

x⃗=ay⃗+bz⃗\vec{x} = a \vec{y} + b \vec{z}

Given vectors:

x⃗=[2583],y⃗=[2−105],z⃗=[−122−3]\vec{x} = \begin{bmatrix} 2 \\ 5 \\ 8 \\ 3 \end{bmatrix}, \quad \vec{y} = \begin{bmatrix} 2 \\ -1 \\ 0 \\ 5 \end{bmatrix}, \quad \vec{z} = \begin{bmatrix} -1 \\ 2 \\ 2 \\ -3 \end{bmatrix}

Substituting the linear combination, we have:

[2583]=a[2−105]+b[−122−3]\begin{bmatrix} 2 \\ 5 \\ 8 \\ 3 \end{bmatrix} = a \begin{bmatrix} 2 \\ -1 \\ 0 \\ 5 \end{bmatrix} + b \begin{bmatrix} -1 \\ 2 \\ 2 \\ -3 \end{bmatrix}

Expanding this equation leads to a system of linear equations:

2 = 2a - b \\ 5 = -a + 2b \\ 8 = 0a + 2b \\ 3 = 5a - 3b \end{cases}$$ Solving this system will allow us to find values of $$a$$ and $$b$$ (if they exist) to confirm that $$\vec{x}$$ is in the span of $$\{\vec{y}, \vec{z}\}$$. ### Solving the System 1. From the third equation, $$8 = 2b$$, we can solve for $$b$$: $$b = \frac{8}{2} = 4$$ 2. Substitute $$b = 4$$ into the first equation: $$2 = 2a - 4 \Rightarrow 2a = 6 \Rightarrow a = 3$$ 3. Substitute $$a = 3$$ and $$b = 4$$ into the remaining equations to verify consistency. - Second equation: $$5 = -3 + 2 \cdot 4 = 5 \quad \text{(True)}$$ - Fourth equation: $$3 = 5 \cdot 3 - 3 \cdot 4 = 3 \quad \text{(True)}$$ ### Conclusion The values $$a = 3$$ and $$b = 4$$ satisfy all four equations, so: $$\vec{x} = 3\vec{y} + 4\vec{z}$$ Thus, $$\vec{x}$$ is indeed in the span of $$\{\vec{y}, \vec{z}\}$$. Would you like further details on any part of the solution? --- ### Related Questions 1. What is the span of a set of vectors, and why is it important in linear algebra? 2. How can we determine if a vector is linearly dependent on other vectors? 3. How would the solution change if one of the components of $$\vec{x}$$, $$\vec{y}$$, or $$\vec{z}$$ were different? 4. Can we use matrix methods (like Gaussian elimination) to solve the system instead? 5. How do the concepts of basis and dimension relate to vector spans? ### Tip When checking if a vector is in the span of others, setting up and solving a system of linear equations is often the most efficient approach.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Span
Linear Combination

Formulas

\(\vec{x} = a\vec{y} + b\vec{z}\)

Theorems

Linear Combination
Span Definition

Suitable Grade Level

College (Introductory Linear Algebra)

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