Problem 25 - Advanced Engineering Mathematics

Question

In Problems \(25-28\), determine whether the given vectors are linearly independent or linearly dependent. $$ \langle 4,-8\rangle,(-6,12) \text { in } R^{2} $$

Step-by-Step Solution

Answer

The vectors are linearly dependent.

Step 1: Define Linear Dependence

Vectors \( \mathbf{v_1} = \langle 4, -8 \rangle \) and \( \mathbf{v_2} = \langle -6, 12 \rangle \) in \( \mathbb{R}^2 \) are said to be linearly dependent if there exist scalars \( c_1 \) and \( c_2 \), not both zero, such that \( c_1 \mathbf{v_1} + c_2 \mathbf{v_2} = \mathbf{0} \). Alternatively, they are linearly independent if the only solution is \( c_1 = c_2 = 0 \).

Step 2: Set Up the Equation

Write the linear combination equation as \( c_1 \langle 4, -8 \rangle + c_2 \langle -6, 12 \rangle = \langle 0, 0 \rangle \). This translates into a system of equations: \(4c_1 - 6c_2 = 0\) and \(-8c_1 + 12c_2 = 0\).

Step 3: Solve the System of Equations

Consider the first equation, \(4c_1 - 6c_2 = 0\), which simplifies to \(2c_1 - 3c_2 = 0\). This implies \(2c_1 = 3c_2\) or \(c_1 = \frac{3}{2}c_2\). Substitute \(c_1 = \frac{3}{2}c_2\) into the second equation: \(-8(\frac{3}{2}c_2) + 12c_2 = 0\), simplifying to \(-12c_2 + 12c_2 = 0\), which is true for any \(c_2\).

Step 4: Conclusion on Dependence

Since we found a non-trivial solution \(c_1 = \frac{3}{2}c_2 \), where \( c_2 eq 0 \) can be chosen, the vectors \( \langle 4, -8 \rangle \) and \( \langle -6, 12 \rangle \) are linearly dependent.

Key Concepts

Vectors

Vectors are fundamental elements in mathematics that have both magnitude and direction. In more practical terms, think of vectors as arrows pointing in space. They are usually depicted in a coordinate system: using two numbers in two-dimensional space like the pair \( \langle 4, -8 \rangle \).

Vectors serve various purposes such as representing quantities having direction and magnitude, like velocity or force. Here are a few essential features of vectors:

• They can be added together to produce a new vector.
• They can be scaled (multiplied by a scalar, which is just a regular number) to change their magnitude.
• They can define a space's dimension, such as two vectors spanning a plane.

If vectors lie in a plane, as in this scenario with \( \mathbb{R}^2 \), you primarily deal with vectors in two dimensions. Vectors play a crucial role in determining whether sets are linearly dependent or independent.
Understanding vectors thoroughly simplifies grasping more complex topics like linear dependence and systems of equations.

Linear Independence

The concept of linear independence is essential when working with vectors. When we say vectors are linearly independent, we mean there's only the trivial solution to their linear combination (where all scalar coefficients are zero) equating to the zero vector. Let's break it down:

• Suppose you have a set of vectors such as \( \langle 4,-8 \rangle \) and \( \langle -6,12 \rangle \).
• They are linearly dependent if you can find non-zero scalars, \(c_1\) and \(c_2\), such that \(c_1 \mathbf{v_1} + c_2 \mathbf{v_2} = \mathbf{0}\).
• Alternatively, they are independent if the only solution is \(c_1 = c_2 = 0\).

In simpler terms, linear dependence implies that the vectors are collinear or in the same line when plotted, so one vector can be expressed as a scalar multiple of the other.
When vectors are independent, they offer a unique set of directions in space, and none is redundant, meaning they span the space uniquely without overlap. This property is crucial in various fields including physics, computer graphics, and more.

Systems of Equations

Systems of equations are collections of two or more equations with a set of variables. Solving them involves finding values for variables that satisfy all equations simultaneously. For instance, consider the system derived from vector equations:

• First equation: \(4c_1 - 6c_2 = 0\).
• Second equation: \(-8c_1 + 12c_2 = 0\).

These equations reflect the conditions needed for the vectors to construct a zero vector—important for checking linear dependence.

To solve such systems, methods like substitution or elimination are employed to derive solutions for variables \(c_1\) and \(c_2\). This leads to discovering relationships between them, such as finding \(c_1 = \frac{3}{2}c_2\), indicating they are dependent.

Add these concepts together, and you see how systems of equations are valuable tools in evaluating vector relationships and underpin many practical applications.