VEC-0010: Introduction to Vectors
We introduce vectors and notation associated with vectors in standard position.
VEC-0030: Vector Arithmetic
We define vector addition and scalar multiplication algebraically and geometrically.
VEC-0035: Standard Unit Vectors in ℝn
We introduce standard unit vectors in , and , and express a given vector as a linear combination of standard unit vectors.
VEC-0040: Linear Combinations of Vectors
We define a linear combination of vectors and examine whether a given vector may be expressed as a linear combination of other vectors, both algebraically and geometrically.
VEC-0050: Dot Product and its Properties
We define the dot product and prove its algebraic properties.
VEC-0060: Dot Product and the Angle Between Vectors
We state and prove the cosine formula for the dot product of two vectors, and show that two vectors are orthogonal if and only if their dot product is zero.
VEC-0070: Orthogonal Projections
We find the projection of a vector onto a given non-zero vector, and find the distance between a point and a line.
RRN-0030: Planes in ℝ3
We establish that a plane is determined by a point and a normal vector, and use this information to derive a general equation for planes in .
SYS-0010: Introduction to Systems of Linear Equations
We solve systems of equations in two and three variables and interpret the results geometrically.
SYS-0020: Augmented Matrix Notation and Elementary Row Operations
We introduce the augmented matrix notation and solve linear system by carrying augmented matrices to row-echelon or reduced row-echelon form.
SYS-0030: Gaussian Elimination and Rank
We introduce Gaussian elimination and Gauss-Jordan elimination algorithms, and define the rank of a matrix.
MAT-0010: Addition and Scalar Multiplication of Matrices
We introduce matrices, define matrix addition and scalar multiplication, and prove properties of those operations.
MAT-0020: Matrix Multiplication
We introduce matrix-vector and matrix-matrix multiplication, and interpret matrix-vector multiplication as linear combination of the columns of the matrix.
MAT-0025: Transpose of a Matrix
We define the transpose of a matrix and state several properties of the transpose. We introduce symmetric, skew symmetric and diagonal matrices.
MAT-0030: Linear Systems as Matrix and Linear Combination Equations
We interpret linear systems as matrix equations and as equations involving linear combinations of vectors. We define singular and nonsingular matrices.
We define the span of a collection of vectors and explore the concept algebraically and geometrically.
VEC-0100: Linear Independence
We define linear independence of a set of vectors, and explore this concept algebraically and geometrically.
SYS-0050: Homogeneous Linear Systems
We define a homogeneous linear system and express a solution to a system of equations as a sum of a particular solution and the general solution to the associated homogeneous system.
MAT-0050: The Inverse of a Matrix
We develop a method for finding the inverse of a square matrix, discuss when the inverse does not exist, and use matrix inverses to solve matrix equations.
MAT-0060: Elementary Matrices
We introduce elementary matrices and demonstrate how multiplication of a matrix by an elementary matrix is equivalent to to performing an elementary row operation on the matrix.
VEC-0110: Linear Independence and Matrices
We prove several results concerning linear independence of rows and columns of a matrix.
VSP-0020: ℝn and Subspaces of ℝn
We define closure under addition and scalar multiplication, and we demonstrate how to determine whether a subset of vectors in is a subspace of .
VSP-0035: Bases and Dimension
We discuss existence of bases of and subspaces of , and define dimension.
VSP-0040: Subspaces of ℝn Associated with Matrices
We define the row space, the column space, and the null space of a matrix, and we prove the Rank-Nullity Theorem.
VSP-0050: Abstract Vector Spaces
We state the definition of an abstract vector space, and learn how to determine if a given set with two operations is a vector space. We define a subspace of a vector space and state the subspace test. We find linear combinations and span of elements of a vector space.
VSP-0060: Bases and Dimension for Abstract Vector Spaces
We revisit the definitions of linear independence, bases, and dimension in the context of abstract vector spaces.
LTR-0010: Introduction to Linear Transformations
We define a linear transformation from into and determine whether a given transformation is linear.
LTR-0020: Standard Matrix of a Linear Transformation from ℝn to ℝm
We establish that every linear transformation of is a matrix transformation, and define the standard matrix of a linear transformation.
LTR-0022: Linear Transformations of Abstract Vector Spaces
We define linear transformation for abstract vector spaces, and illustrate the definition with examples.
LTR-0025: Linear Transformations and Bases
We establish that a linear transformation of a vector space is completely determined by its action on a basis.
LTR-0030: Composition and Inverses of Linear Transformations
We define composition of linear transformations, inverse of a linear transformation, and discuss existence and uniqueness of inverses.
LTR-0035: Existence of the Inverse of a Linear Transformation
We prove that a linear transformation has an inverse if and only if the transformation is “one-to-one” and “onto”.
LTR-0070: Geometric Transformations of the Plane
We find standard matrices for classic transformations of the plane such as scalings, shears, rotations and reflections.
LTR-0050: Image and Kernel of a Linear Transformation
We define the image and kernel of a linear transformation and prove the Rank-Nullity Theorem for linear transformations.
LTR-0060: Isomorphic Vector Spaces
We define isomorphic vector spaces, discuss isomorphisms and their properties, and prove that any vector space of dimension is isomorphic to .
LTR-0080: Matrix of a Linear Transformation with Respect to Arbitrary Bases
We find the matrix of a linear transformation with respect to arbitrary bases, and find the matrix of an inverse linear transformation.
DET-0010: Definition of the Determinant – Expansion Along the First Row
We define the determinant of a square matrix in terms of cofactor expansion along the first row.
DET-0020: Definition of the Determinant – Expansion Along the First Column
We define the determinant of a square matrix in terms of cofactor expansion along the first column, and show that this definition is equivalent to the definition in terms of cofactor expansion along the first row.
DET-0030: Elementary Row Operations and the Determinant
We examine the effect of elementary row operations on the determinant and use row reduction algorithm to compute the determinant.
DET-0040: Properties of the Determinant
We summarize the properties of the determinant that we already proved, and prove that a matrix is singular if and only if its determinant is zero, the determinant of a product is the product of the determinants, and the determinant of the transpose is equal to the determinant of the matrix.
DET-0050: The Laplace Expansion Theorem
We state and prove the Laplace Expansion Theorem for determinants.
DET-0060: Determinants and Inverses of Nonsingular Matrices
We derive the formula for Cramer’s rule and use it to express the inverse of a matrix in terms of determinants.
VEC-0080: Cross Product and its Properties
We define the cross product and prove several algebraic and geometric properties.
DET-0070: Determinants as Areas and Volumes
We interpret a determinant as the area of a parallelogram, and a determinant as the volume of a parallelepiped.
EIG-0010: Describing Eigenvalues and Eigenvectors Algebraically and Geometrically
We introduce the concepts of eigenvalues and eigenvectors of a matrix.
EIG-0020: Finding Eigenvalues and Eigenvectors
We explore the theory behind finding the eigenvalues and associated eigenvectors of a square matrix.
EIG-0050: Diagonalizable Matrices and Multiplicity
In this module we discuss algebraic multiplicity, geometric multiplicity, and their relationship to diagonalizability.