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Recent Notes

March 13, 2018

Second Midterm, Copy 1

Elementary Calculus Second Midterm, Copy 1 1. Differentiate (1)   Solution: We know (2)   We can use product rule to write (3)     We […]
February 8, 2018

Elementary Calculus, First Midterm, Copy 2

Elementary Calculus, First Midterm, Copy 2 1. Suppose is a given function and and are real numbers. State what we mean by (1)     Solution: […]
February 8, 2018

Elementary Calculus, First Midterm, Copy 1

Elementary Calculus, First Midterm, Copy 1 1. Suppose is a given function and is a real number. State what we mean by (1)     Solution: […]
November 16, 2017

Power Series

Example 1.   We want to find all numbers for which the power series (1)   converges. Define for every and use Ratio test to write […]
November 16, 2017

Eigenvalues and Eigenvectors

Example 1.   Suppose we are given the map defined by . We would like to check if is linear. Take two vectors and and write […]
November 12, 2017

Gram-Schmidt Process

Gram-Schmidt Process, Orthonormal Basis Example 1.   We are given the set . First we notice that (1)   and therefore the two vectors in this […]
November 2, 2017

Rank and Nullity

Rank and Nullity Example 1.   We want to find a basis for the row space of (1)   We know the row space of is […]
October 9, 2017

Linearly Independent Sets and Basis

Linearly Independent Sets and Basis Example 1.   Suppose we are given the set . We would like to determine if this set is linearly independent. […]
October 3, 2017

Vector Space

Vector Space Example 1. We are given a vector where and are real numbers. We would like to write as a linear combination of vectors and […]
September 28, 2017

More on Determinants

More on Determinants Observation 1.   We are given two matrices and . Assume that matrix is elementary.   Case 1. is obtained from by interchanging […]
September 21, 2017

Determinants

Example 1. We would like to find the determinant of (1)   We write (2)     Example 2. We would like to find the determinant […]
September 6, 2017

Elementary Matrices

Example 1. Suppose we want to find a sequence of elementary matrices that can be used to write (1)   in row-echelon form. We know an […]