Problem 4 - Technical Mathematics with Calculus

Question

A=\(\left(\begin{array}{lll}2 & 5 & 1 \\ 6 & 3 & 7 \\ 1 & 6 & 9 \\ 7 & 4 & 2\end{array}\right)\) B=\(\left(\begin{array}{l}7 \\ 3 \\ 9 \\ 2\end{array}\right)\) C=\(\left(\begin{array}{cccc}f & i & q & w \\ & g & w & k \\ & & c & z \\ & & & b\end{array}\right)\) D=\(\left(\begin{array}{llll}6 & 2 & 0 & 1 \\ 2 & 8 & 3 & 9\end{array}\right)\) E=\(\left(\begin{array}{ll}x & y \\ z & w\end{array}\right)\) F=\(\left(\begin{array}{llll}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right)\) G=7 H=\(\left(\begin{array}{cc}3 & 8 \\ & 5\end{array}\right)\) I=\(\left(\begin{array}{llll}3 & 8 & 4 & 6\end{array}\right)\) J=\(\left(\begin{array}{llll}3 & 7 & 2 & 1 \\ 5 & 2 & 9 & 3 \\ 5 & 1 & 7 & 2 \\\ 7 & 3 & 9 & 1\end{array}\right)\) K=\(\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1\end{array}\right)\) Which of the 11 arrays shown is a column vector?

Step-by-Step Solution

Answer

B is a column vector.

Step 1: Examine the Problem

A=\(\left(\begin{array}{lll}2 & 5 & 1 \\ 6 & 3 & 7 \\ 1 & 6 & 9 \\ 7 & 4 &
2\end{array}\right)\)
B=\(\left(\begin{array}{l}7 \\ 3 \\ 9 \\ 2\end{array}\right)\)
C=\(\left(\begin{array}{cccc}f & i & q & w \\ & g & w & k \\ & & c & z \\ &
& & b\end{array}\right)\)
D=\(\left(\begin{array}{llll}6 &

Step 2: Apply Mathematical Methods

We use the relevant mathematical definitions and methods to work through the problem.

Step 3: Answer

B is a column vector.

Key Concepts

Matrix

A matrix is a rectangular arrangement of numbers, symbols, or expressions, set out in rows and columns, which is used in mathematics, especially in linear algebra, to model and solve complex problems. The individual items within a matrix are called elements. In linear algebra, matrices can represent systems of linear equations, transformations in space, and can be used in areas like physics, economics, statistics, and computer science.

Matrices come in various sizes, defined by the number of rows and columns they have. When talking about the size of a matrix, we usually specify the number of rows first, followed by the number of columns. A matrix with the same number of rows and columns is called a square matrix, like matrix K in the original exercise, which is a 4x4 square matrix.

Array

In the context of linear algebra, an array is a systematic arrangement of numbers or variables in rows and columns, similar to a matrix. However, the term can be used more broadly in computer science to refer to a data structure that can store multiple values. For our purposes, we use 'array' and 'matrix' interchangeably.

An important special case of an array is the column vector, which is essentially a matrix with just one column, such as B in the original exercise. Column vectors are vital in linear algebra because they can represent vectors in multidimensional space, each row corresponding to a different dimension's magnitude.

Linear Algebra

Linear algebra is a branch of mathematics concerned with vectors, vector spaces, linear mappings, and systems of linear equations. It provides a way to solve for unknowns that are linearly related and includes the study of lines, planes, and subspaces. Matrices and arrays are foundational tools used in linear algebra to represent these linear transformations and to perform operations such as addition, subtraction, and multiplication.

Identifying structures, such as column vectors or square matrices, and understanding their properties are crucial skills in linear algebra. For example, the solution of identifying a column vector within a series of matrices or arrays is fundamental because column vectors represent points in space or the coefficients of systems of linear equations.