Final answer:
To determine if vectors are linearly independent by inspection, one checks if any vector is a multiple of another. In two-dimensional vector problems, we analyze the independence of horizontal and vertical components. For data sets, we use scatter plots and regression to assess linear correlation and perform tests of independence for categorical variables.
Step-by-step explanation:
To determine by inspection whether vectors are linearly independent, you first need to look at the given vectors and check for any obvious relationships between them. If one vector is a multiple of another, or if a vector can be written as a combination of other vectors in the set, then the vectors are linearly dependent.
In the context of two-dimensional motion, horizontal and vertical vectors are independent if no horizontal vector can be written as a multiple of a vertical vector and vice versa. For mathematical problems involving vectors, it's often useful to pick a coordinate system and project the vectors onto the x and y axes, then use trigonometric functions to find their components.
When assessing data sets for linear correlation, we typically use scatter plots and regression analysis to find the line of best fit and calculate the correlation coefficient to determine the strength and significance of the relationship between variables.
A test of independence in probability and statistics is used to determine if there is a significant relationship between two categorical variables. This often involves calculating expected frequencies and comparing them with observed frequencies in a contingency table. In vector analysis, independence also has a specific meaning, describing a scenario where no single vector in a set can be represented as a linear combination of others.