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One may get confused when asked to specify the coordinates of T as depicted below. Figure 1 However, they may think it easy to determine the coordinates of T if instead of Figure 1, they are given the following figure.Read More
Just as the coordinate matrix of a vector depends on the basis chosen for the associated vector space, the matrix of a linear operator on a vector space depends on the basis for the vector space. It is often favorableRead More
Consider the matrix , which can be viewed as a matrix representation of a linear operator on , and the vector . It is easy to check that . In this example, the output of the matrix transformation is aRead More
Problem 1 Find the population principal components Y1 and Y2 for the covariance matrix . Then, calculate the proportion of the total population variance explained by the first principal component. Answer Find the eigenvalues and the corresponding eigenvectors. (λ-5)(λ-2) –Read More
Part 1 of this article introduces the concept of arithmetic sequence. Like sequences in general, arithmetic sequences can generate what we call arithmetic series. In brief, for every term of an arithmetic sequence there corresponds a partial sum; the sequenceRead More
Consider the following sequences of real numbers. Sequence 1: 2, 5, 8, 11, 14, … Sequence 2: 11, 16, 21, 26, 31, … Sequence 3: 15, 13, 11, 9, 7, … They are examples of what weRead More
The mean-value theorem is one of the most important theorems in calculus. However, despite its roles in the development of calculus itself, the mean-value theorem can be used in estimation-related problems. This article provides some examples of how it isRead More
In the article Quadratic Functions, we infer the properties of quadratic functions based on given equations of the form f(x) = ax2 + bx + c where a ≠ 0. In the opposite direction, this article describes how to findRead More
Just as the concept of one-sided limits in the study of limits of functions, we may find the concept of one-sided derivatives when we learn calculus. In the article The Limit of A Function at A Point, it is claimedRead More
Given the lengths of the sides of a triangle, how to find the lengths of the medians? There are several formulae for determining the lengths of the medians. They are easily derived by applying Stewart’s Theorem. Stewart’s Theorem Let ABCRead More