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Recent Notes

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 […]
August 21, 2017

Matrix Inverse

Example 1. We want to find inverse of the matrix (1)   This means we want to find a matrix such that (2)   We start […]
August 18, 2017

Matrix & Basic Operations

Suppose (1)   and we would like to compute when possible.   Let’s begin with the first one (2)     Next, we know multiplying two […]
August 12, 2017

Linear System and Matrix

Example 1. We are given the system (1)   The matrix associated to this system is (2)   We want to find the solution set of […]