Project and contact information.

We define $\RR ^n$ and learn how to plot points in $\RR ^3$.

We introduce vectors and notation associated with vectors in standard position.

We find vector magnitude.

We define vector addition and scalar multiplication algebraically and geometrically.

We introduce standard unit vectors in $\RR ^2$, $\RR ^3$ and $\RR ^n$, and express a given vector as a linear combination of standard unit 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.

We define the dot product and prove its algebraic properties.

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.

We find the projection of a vector onto a given non-zero vector, and find the distance between a point and a line.

We use parametric equations to represent lines in $\RR ^2$, $\RR ^3$ and $\RR ^n$.

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 $\RR ^3$.

We solve systems of equations in two and three variables and interpret the results geometrically.

We introduce the augmented matrix notation and solve linear system by carrying augmented matrices to row-echelon or reduced row-echelon form.

We introduce Gaussian elimination and Gauss-Jordan elimination algorithms, and define the rank of a matrix.

We introduce matrices, define matrix addition and scalar multiplication, and prove properties of those operations.

We introduce matrix-vector and matrix-matrix multiplication, and interpret matrix-vector multiplication as linear combination of the columns of the matrix.

We present and practice block matrix multiplication.

We define the transpose of a matrix and state several properties of the transpose. We introduce symmetric, skew symmetric and diagonal matrices.

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.

We define linear independence of a set of vectors, and explore this concept algebraically and geometrically.

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.

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.

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.

We prove several results concerning linear independence of rows and columns of a matrix.

We define closure under addition and scalar multiplication, and we demonstrate how to determine whether a subset of vectors in $\RR ^n$ is a subspace of $\RR ^n$.

We define bases and consider examples of bases of $\RR ^n$ and subspaces of $\RR ^n$.

We discuss existence of bases of $\RR ^n$ and subspaces of $\RR ^n$, and define dimension.

We define the row space, the column space, and the null space of a matrix, and we prove the Rank-Nullity Theorem.

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.

We revisit the definitions of linear independence, bases, and dimension in the context of abstract vector spaces.

We define a linear transformation from $\RR ^n$ into $\RR ^m$ and determine whether a given transformation is linear.

We establish that every linear transformation of $\RR ^n$ is a matrix transformation, and define the standard matrix of a linear transformation.

We define linear transformation for abstract vector spaces, and illustrate the definition with examples.

We establish that a linear transformation of a vector space is completely determined by its action on a basis.

We define composition of linear transformations, inverse of a linear transformation, and discuss existence and uniqueness of inverses.

We prove that a linear transformation has an inverse if and only if the transformation is “one-to-one” and “onto”.

We find standard matrices for classic transformations of the plane such as scalings, shears, rotations and reflections.

We define the image and kernel of a linear transformation and prove the Rank-Nullity Theorem for linear transformations.

We define isomorphic vector spaces, discuss isomorphisms and their properties, and prove that any vector space of dimension $n$ is isomorphic to $\RR ^n$.

We find the matrix of a linear transformation with respect to arbitrary bases, and find the matrix of an inverse linear transformation.

We define the determinant of a square matrix in terms of cofactor expansion along the first row.

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.

We examine the effect of elementary row operations on the determinant and use row reduction algorithm to compute 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.

We state and prove the Laplace Expansion Theorem for determinants.

We derive the formula for Cramer’s rule and use it to express the inverse of a matrix in terms of determinants.

We define the cross product and prove several algebraic and geometric properties.

We interpret a $2\times 2$ determinant as the area of a parallelogram, and a $3\times 3$ determinant as the volume of a parallelepiped.

We introduce the concepts of eigenvalues and eigenvectors of a matrix.

We explore the theory behind finding the eigenvalues and associated eigenvectors of a square matrix.

Abstract goes here

In this module we discuss algebraic multiplicity, geometric multiplicity, and their relationship to diagonalizability.

We introduce the augmented matrix notation and solve linear system by carrying augmented matrices to row-echelon or reduced row-echelon form.