Problem 24 - Advanced Engineering Mathematics

Question

Write the given sum as a single-column matrix. $$ 4\left(\begin{array}{r} -1 \\ 2 \end{array}\right)-2\left(\begin{array}{l} 2 \\ 8 \end{array}\right)+3\left(\begin{array}{r} -2 \\ 3 \end{array}\right) $$

Step-by-Step Solution

Answer

The sum as a single-column matrix is \(\begin{array}{r} -14 \\ 1 \end{array}\).

Step 1: Scalar Multiplication

Multiply each vector by its respective scalar. The vector \(4 \left(\begin{array}{r} -1 \ 2 \end{array}\right)\) becomes \(\left(\begin{array}{r} 4(-1) \ 4(2) \end{array}\right) = \left(\begin{array}{r} -4 \ 8 \end{array}\right)\). The vector \(-2 \left(\begin{array}{l} 2 \ 8 \end{array}\right)\) becomes \(\left(\begin{array}{l} -2(2) \ -2(8) \end{array}\right) = \left(\begin{array}{r} -4 \ -16 \end{array}\right)\). The vector \(3 \left(\begin{array}{r} -2 \ 3 \end{array}\right)\) becomes \(\left(\begin{array}{r} 3(-2) \ 3(3) \end{array}\right) = \left(\begin{array}{r} -6 \ 9 \end{array}\right)\).

Step 2: Sum the Vectors

Add the resulting vectors from Step 1 together. Sum the corresponding elements: \(\left(\begin{array}{r} -4 \ 8 \end{array}\right) + \left(\begin{array}{r} -4 \ -16 \end{array}\right) + \left(\begin{array}{r} -6 \ 9 \end{array}\right)\). For the first element: \(-4 + (-4) + (-6) = -14\). For the second element: \(8 + (-16) + 9 = 1\).

Step 3: Write as a Single-Column Matrix

Put the summed values together to form the final single-column matrix: \(\begin{array}{r} -14 \ 1 \end{array}\).

Key Concepts

Scalar Multiplication

Scalar multiplication is a foundational operation in matrix algebra. It involves multiplying each element of a vector or matrix by a scalar value. This process scales the entire vector or matrix:

• Multiply each entry of a vector or matrix by the scalar.
• The resulting vector or matrix has the same dimensions as the original.
• This operation effectively stretches or compresses the vector or matrix.

For example, let the scalar be 3 and the vector be \(\begin{pmatrix} 2 \ 5 \end{pmatrix}\). The scalar multiplication results in \(3 \times \begin{pmatrix} 2 \ 5 \end{pmatrix} = \begin{pmatrix} 6 \ 15 \end{pmatrix}\). Notice how each component of the vector has been multiplied by 3.
Scalar multiplication is crucial for performing operations like the ones seen in linear differential equations and transformations in graphics.

Vector Addition

Vector addition is another vital operation in linear algebra, used frequently alongside scalar multiplication. In vector addition:

• Align vectors of the same size.
• Add corresponding components to form a new vector.
• This operation can be visualized as placing vectors head-to-tail in a coordinate space.

For example, consider the vectors \(\begin{pmatrix} 1 \ 3 \end{pmatrix}\) and \(\begin{pmatrix} 4 \ -2 \end{pmatrix}\). The resulting vector from their addition is \(\begin{pmatrix} 1+4 \ 3+(-2) \end{pmatrix} = \begin{pmatrix} 5 \ 1 \end{pmatrix}\).
This operation is essential in physics for calculating resultant forces and in computer graphics for manipulating points and models.

Column Matrix

A column matrix, or column vector, is a matrix with only one column and multiple rows. It looks like this:

• Helps represent components of a vector.
• Written as \(\begin{pmatrix} a \ b \ c \end{pmatrix}\) for a three-element vector.
• It's versatile in math and physics for expressing various concepts such as velocities or forces.

The transformation of all exercises in this context usually results in a column matrix. Given an operation involving column matrices, the goal is to ensure results are also in column vector form. Using our initial example, after consolidating operations, the sum’s result \(\begin{pmatrix} -14 \ 1 \end{pmatrix}\) is a column matrix.

Linear Algebra

Linear algebra is the branch of mathematics focused on vectors and matrices, alongside operations like scalar multiplication and vector addition. Core ideas include:

• Solving systems of linear equations.
• Studying spaces called vector spaces.
• Understanding transformations and eigenvectors.

These concepts aid in modeling real-world phenomena, from computer graphics transformations to dynamics in physical systems. In our exercise, linear algebra principles are applied to manipulate and simplify vector expressions using these operations:

1. Scalar multiplication is used to adjust vectors' magnitudes as needed.
2. Vector addition is performed to combine those vectors into one result.
3. The result is organized into a column matrix, demonstrating both structure and simplicity.

With its versatility, linear algebra is essential not just in mathematics but also in fields such as engineering, physics, computer science, and economics.