Question

Assume that array arrayl is defined as shown, and determine the contents of the following sub-arrays. \(\operatorname{array} 1=\left[\begin{array}{rrrrr}1.1 & 0.0 & 2.1 & -3.5 & 6.0 \\\ 0.0 & 1.1 & -6.6 & 2.8 & 3.4 \\ 2.1 & 0.1 & 0.3 & -0.4 & 1.3 \\ -1.4 & 5.2 & 0.0 & 1.1 & 0.0\end{array}\right]\) a. array \(1(3, t)\) b. array \(1(:, 3)\) c. array \(1\left(1: 2: 3,\left[\begin{array}{lll}3 & 3 & 4\end{array}\right]\right)\) d. array \(\left.1\left(\begin{array}{ll}1 & 1\end{array}\right], 4\right)\)

Short answer

a. \[ \left[ \begin{array}{rrrrr} 2.1 & 0.1 & 0.3 & -0.4 & 1.3 \end{array} \right] \] b. \[ \left[ \begin{array}{r} 2.1 \\ -6.6 \\ 0.3 \\ 0.0 \end{array} \right] \] c. \[ \left[ \begin{array}{rr} 2.1 & -3.5 \\ 0.3 & -0.4 \end{array} \right] \] d. \[ -3.5 \]

Step-by-step solution

a. array \(1(3, t)\): Elements from row 3

To find the sub-array corresponding to row 3, look at the third row of array1 and extract all its elements: \[ \left[ \begin{array}{rrrrr} 2.1 & 0.1 & 0.3 & -0.4 & 1.3 \end{array} \right] \]

b. array \(1(:, 3)\): Elements from column 3

To find the sub-array corresponding to column 3, look at the third column of array1 and extract all its elements: \[ \left[ \begin{array}{r} 2.1 \\ -6.6 \\ 0.3 \\ 0.0 \end{array} \right] \]

c. array \(1(1: 2: 3,\left[\begin{array}{lll}3 & 3 & 4\end{array}\right])\): Elements from every second row (1,3) and columns 3 and 4

To find the sub-array with every second row (1,3) and columns 3 and 4, identify the corresponding elements in array1: \[ \left[ \begin{array}{rr} 2.1 & -3.5 \\ 0.3 & -0.4 \end{array} \right] \]

d. array \(\left.1\left(\begin{array}{ll}1 & 1\end{array}\right], 4):\) Elements from row 1, and column 4

To find the sub-array corresponding to row 1, and column 4, identify the corresponding element in array1: \[ -3.5 \]

Key concepts

Sub-array Extraction in MATLAB

Extracting sub-arrays in MATLAB is a vital operation when working with matrices. To visualize, imagine a spreadsheet with rows and columns, and you wish to select certain cells or blocks from it. That's precisely what sub-array extraction in MATLAB allows you to do - select specific elements or groups of elements from a larger array. This is an important technique for analyzing data or creating new arrays from existing ones.

For instance, to extract a single row from a matrix, as in the exercise part (a), you use array1(3, :). This syntax tells MATLAB to select all elements (denoted by the colon :) that are in the third row. Conversely, to extract a single column, such as in part (b), the syntax array1(:, 3) is used where the colon now represents all rows, and the number 3 specifies the third column.

In more advanced sub-array extractions, such as in part (c), you can use colon notation combined with vector indices to extract elements irregularly. Here, array1(1:2:3, [3, 3, 4]) selects every second row starting from the first, and repeats the third column twice before selecting the fourth. Understanding how to manipulate these indices is key to extracting precisely what you need from your dataset.

Matrix Operations Fundamentals

In addition to sub-array extraction, matrix operations are a cornerstone in MATLAB programming. These operations can range from simple arithmetic calculations to more complex manipulations like matrix inversion, eigenvalue computation, or singular value decompositions.

Basic operations include addition, subtraction, and multiplication. MATLAB treats arrays not only as sets of values but also as mathematical objects. For instance, when you multiply two matrices, you are performing a dot product, combining rows of the first matrix with columns of the second (provided they have compatible dimensions). It’s essential to consider this when executing operations, to prevent dimension mismatch errors.

Element-wise operations can be performed too, using a dot (.) before the operator, such as .*, ./, or .^. These are important when you want to apply the operation to each corresponding element of the matrix independently. A deep understanding of these matrix operations allows for robust analysis and manipulation of data within MATLAB.

Understanding MATLAB Syntax

The MATLAB syntax might be daunting at first, but it follows a logical pattern which, once learned, provides a powerful toolset for numerical computations. Properly understanding the MATLAB syntax is imperative for successful programming.

The syntax used for array indexing is a good starting point. Using parentheses, (), indicates that you’re referencing elements within an array. The colon, :, is a powerful operator in MATLAB used to define ranges or to represent entire dimensions. Square brackets, [], denote the creation of arrays or matrices and can also help in defining the indices for sub-array extraction.

Moreover, understanding the interplay between semicolons and commas is crucial. A semicolon (;) is used to separate rows in an array, while a comma (,) separates elements within a row. When combining these syntax elements effectively, you can write concise and robust MATLAB code to perform complex operations, such as those seen in the provided exercise, parts (c) and (d), which combine slicing with array creation.