Show that no one-to-one linear transformation can exist from a3 a 2 . Generalize this result by proving that no one-to-one linear transformation can exist from U V if the dimension of U is greater than the dimension of V.

Organizing and Summarizing Data Andrew Isbell *Organizing Qualitative data: Data which is not yet organized is referred to as “Raw Data” Frequency distribution the frequency of each category and the number of times this category occurs. Relative frequency the proportion/percent of observations within a category ad is found using the formula, Frequency/Sum of all frequencies. Pareto chart a bar graph in which the bars are in decreasing order, from tallest to shortest Pie chart a circle divided into sections which each represents a category of data by utilizing relative frequency Bar graphs display either frequency or relative frequency while pie charts can only display relative frequency