Problem 26 - Advanced Engineering Mathematics

Question

In Problems \(25-28\), determine whether the given vectors are linearly independent or linearly dependent. $$ \langle 1,1\rangle,\langle 0,1\rangle,\langle 2,5\rangle \text { in } R^{2} $$

Step-by-Step Solution

Answer

The vectors \( \langle 1,1 \rangle, \langle 0,1 \rangle, \langle 2,5 \rangle \) are linearly dependent.

Step 1: Understand the Concept

Linear independence of vectors means that no vector in the set can be written as a linear combination of the others. If vectors are linearly dependent, at least one can be expressed as a combination of the others. Check if these vectors are independent.

Step 2: Set Up the Linear Combination

Formulate the equation for linear dependence: If \\( a\langle 1,1 \rangle + b\langle 0,1 \rangle + c\langle 2,5 \rangle = \langle 0,0 \rangle \) \for coefficients \( a, b, \) and \( c \), not all zero, the vectors are dependent.

Step 3: Write and Simplify the Equations

The above equation gives two equations: \\( a + 2c = 0 \) and \\( a + b + 5c = 0 \). Simplifying this system helps determine if non-zero solutions exist for \( a, b, \) and \( c \).

Step 4: Solve the System of Equations

First equation \( a + 2c = 0 \) shows \( a = -2c \). Substitute into the second equation: \(-2c + b + 5c = 0\). Simplifying: \( b + 3c = 0 \). This means \( b = -3c \).

Step 5: Determine Coefficients

Assigning a value to \( c \) (e.g., \( c = 1 \)), we find \( a = -2, b = -3 \). Since \( a, b, c \) are not all zeros, the vectors are linearly dependent.

Key Concepts

Vector Algebra

Vector algebra forms the backbone of linear dependence and independence analysis. At its core, vector algebra involves understanding operations like vector addition and scalar multiplication.
These operations allow you to manipulate vectors in a space such as \( \mathbb{R}^2 \), where you combine vectors or scale them with numbers. Consider the vectors given in the problem: \( \langle 1,1 \rangle, \langle 0,1 \rangle, \langle 2,5 \rangle \).
In this space, you can perform operations like:

• Adding vectors: Combine vectors by adding their corresponding components.
• Scalar multiplication: Multiply a vector by a real number (scalar), scaling each component.

Combining these techniques, vector algebra sets the stage for complex operations like forming linear combinations, which are essential in determining linear dependence.

Systems of Equations

To determine if vectors are linearly dependent, we use a system of equations derived from linear combinations. This approach is essential for unraveling the relationships between vectors. The equations are derived from the statement \( a\langle 1,1 \rangle + b\langle 0,1 \rangle + c\langle 2,5 \rangle = \langle 0,0 \rangle \).
Breaking this into systems, we get:

• First equation: \( a + 2c = 0 \)
• Second equation: \( a + b + 5c = 0 \)

These equations are simplified and solved to find values for \( a, b, \) and \( c \).
Solving involves expressing variables in terms of one another to determine possible solutions. Systems of equations are pivotal for translating the algebraic conditions of dependence into explicit numerical relationships.
Here, this system shows that non-zero values exist for these coefficients, hence the vectors are linearly dependent.

Linear Combinations

Linear combinations synthesize multiple vectors to evaluate dependence. In this context, a linear combination of vectors involves scaling each vector by a coefficient and adding the results.
This concept is central in assessing whether any vector in the set can be made through these operations on the others. In practice, you do this by verifying if the only solution to \( a\langle 1,1 \rangle + b\langle 0,1 \rangle + c\langle 2,5 \rangle = \langle 0,0 \rangle \) is trivial (all coefficients zero).

• If a non-trivial solution exists, the vectors are dependent.
• Each coefficient represents the 'weight' of a vector in forming another.

By setting these vectors in a linear combination to zero, you determine if one vector can be formed from the others, clearly indicating their dependence.