Problem 4 - Differential Equations and Linear Algebra

Question

If \(\mathbf{x}=(3,-1,2,5)\) and \(\mathbf{y}=(-1,2,9,-2),\) determine \(\mathbf{v}=5 \mathbf{x}+(-7) \mathbf{y}\) and its additive inverse.

Step-by-Step Solution

Answer

The vector \(\mathbf{v}=5\mathbf{x}+(-7)\mathbf{y}\) is \((22, -19, -53, 39)\), and its additive inverse is \((-22, 19, 53, -39)\).

Step 1: Multiply the vectors by their scalars

Multiply the vector \(\mathbf{x}\) by the scalar 5 and the vector \(\mathbf{y}\) by the scalar (-7): \(5\mathbf{x} = 5(3, -1, 2, 5)\) \((-7)\mathbf{y} = -7(-1, 2, 9, -2)\)

Step 2: Calculate the resulting vectors after scalar multiplication

Compute the results of the scalar multiplication: \(5\mathbf{x} = (15, -5, 10, 25)\) \((-7)\mathbf{y} = (7, -14, -63, 14)\)

Step 3: Add the resulting vectors

Add the vectors obtained in Step 2: \(\mathbf{v} = 5\mathbf{x} + (-7)\mathbf{y} = (15, -5, 10, 25) + (7, -14, -63, 14)\)

Step 4: Calculate the resulting vector

Compute the sum of the two vectors: \(\mathbf{v} = (15+7, -5-14, 10-63, 25+14) = (22, -19, -53, 39)\)

Step 5: Find the additive inverse

The additive inverse of a vector is the opposite of that vector. Calculate the additive inverse of \(\mathbf{v}\) by changing the signs of its components: \(-\mathbf{v} = -1(22, -19, -53, 39) = (-22, 19, 53, -39)\) The vector \(\mathbf{v}=5\mathbf{x}+(-7)\mathbf{y}\) is \((22, -19, -53, 39)\), and its additive inverse is \((-22, 19, 53, -39)\).

Key Concepts

Vector Addition

In the realm of mathematics, particularly in vector algebra, vector addition is a fundamental operation that combines two or more vectors to produce a new vector. Think of vectors as arrows in space, each with a direction and length, representing quantities such as force, velocity, or displacement. When we add two vectors, we're essentially putting them nose-to-tail to find their resultant.

For example, if we have two vectors, \(\textbf{a} = (a_1, a_2,..., a_n)\) and \(\textbf{b} = (b_1, b_2,..., b_n)\), where \(n\) represents the dimension of the vectors, their sum \(\textbf{c} = \textbf{a} + \textbf{b}\) is given by adding each corresponding pair of components: \(\textbf{c} = (a_1+b_1, a_2+b_2,..., a_n+b_n)\).

This concept is visually represented by drawing vectors on a graph, then adding them using the aforementioned nose-to-tail method or, as in this specific exercise, simply by adding corresponding components algebraically. This approach is applied when we compute \(5\textbf{x} + (-7)\textbf{y}\) resulting in a new vector \(\textbf{v}\).

Additive Inverse of a Vector

Delving deeper into vector arithmetic, we encounter the notion of an additive inverse. To understand this, consider a simple number line: the negative of any number is its additive inverse because their sum equals zero. Apply this idea to vectors: the additive inverse of a vector \(\textbf{v}\) is another vector which, when added to \(\textbf{v}\), will result in the zero vector (the vector equivalent of zero).

Mathematically, if \(\textbf{v} = (v_1, v_2,..., v_n)\), then its additive inverse is \(-\textbf{v} = (-v_1, -v_2,..., -v_n)\). Obtaining the additive inverse involves simply reversing the signs of all the components in the original vector. In the context of the exercise provided, the additive inverse of vector \(\textbf{v} = (22, -19, -53, 39)\) is \(-\textbf{v} = (-22, 19, 53, -39)\), effectively giving us a vector that 'cancels out' the original when combined.

Linear Combination of Vectors

A crucial operation in vector spaces is called a linear combination. This occurs when we combine several vectors using scalar multiplication (multiplying a vector by a number) and vector addition. The concept can be likened to mixing ingredients in specific proportions to yield a new recipe. Similarly, in linear algebra, we mix vectors with certain 'weights' (scalars) to create new vector 'dishes.'

Specifically, suppose we want to combine vectors \(\textbf{a}\), \(\textbf{b}\), ..., \(\textbf{n}\) using scalars \(c_1, c_2, ..., c_n\). The linear combination would be expressed as \(c_1\textbf{a} + c_2\textbf{b} + ... + c_n\textbf{n}\). This process gives us the tools to explore many topics in mathematics and physics where things can be expressed as a combination of basic building blocks or components.

In our exercise, \(5\textbf{x} + (-7)\textbf{y}\) is a linear combination of vectors \(\textbf{x}\) and \(\textbf{y}\) with scalars 5 and -7, culminating in the vector \(\textbf{v}\).