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 […]

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 […]

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 […]

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 […]

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. […]

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 […]

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 […]

Determinants

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

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 […]