Answered: I want a transformation matrix that I…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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I want a transformation matrix that I can multiply with the matrix to arrive at a solution. An explanation with the answer will be much appreciated.
preparing for a quiz on monday

Expert Solution

Step 1

To scale a matrix by a factor of 'a' along the x-axis and 'b' along the y-axis, we can create a scaling matrix S given by:

$[\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & 1 \end{matrix}]$

We can then multiply the original matrix M with S to get the scaled matrix M' as follows:

M' = S * M

Given

M= [[1,2,1],[3,7,3],[4,5,4]]

a = 2

b = 0.5

For the given matrix M = [[1,2,1],[3,7,3],[4,5,4]], the scaling factors are a = 2 along the x-axis and b = 0.5 along the y-axis. So the scaling matrix S is:

$[\begin{matrix} 2 & 0 & 0 \\ 0 & 0.5 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Multiplying M with S gives:

$[\begin{matrix} 2 & 0 & 0 \\ 0 & 0.5 & 0 \\ 0 & 0 & 1 \end{matrix}]$ * $[\begin{matrix} 1 & 2 & 1 \\ 3 & 7 & 3 \\ 4 & 5 & 4 \end{matrix}]$ = $[\begin{matrix} 2 & 4 & 2 \\ 1 & 3.5 & 1.5 \\ 4 & 2.5 & 4 \end{matrix}]$

So the scaled matrix M' is:

$[\begin{matrix} 2 & 4 & 2 \\ 1 & 3.5 & 1.5 \\ 4 & 2.5 & 4 \end{matrix}]$

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