## Topical Outline with Specific Course Objectives

- Systems of Linear Equations
- Recognize a linear equation in
*n*variables. - Find a parametric representation of a solution set of a system of linear equations.
- Write a system of linear equations in matrix form.
- Solve a system of linear equations by substitution, graphing, using a computer or graphing calculator, Gaussian elimination, Gauss-Jordan elimination, LU-factorization, Cramer’s Rule.
- Determine whether a system of linear equations is consistent or inconsistent.
- Write a given system of linear equations in the form
**Ax=b**and use it to solve for**x**. - Find a general solution of a consistent system.

- Recognize a linear equation in
- Matrices
- Determine the size, transpose, inverse, rank, and LU-factorization of a matrix.
- Write an augmented or coefficient matrix from a system of linear equations.
- Use elementary row operations to write a matrix in row-echelon form and reduced row-echelon form.
- Perform matrix operations and solve matrix equations.
- Factor a given matrix into a product of elementary matrices.
- Find the adjoint of a matrix and use it to find the inverse of the matrix. Know and use equivalent conditions for an invertible matrix.

- Determinants
- Find the determinant, minors and cofactors of a given matrix.
- Use the determinant to decide whether a given matrix is singular or nonsingular.
- Use properties of determinants.

- Vectors and Vector Spaces
- Perform vector operations for vectors in R
^{n}. - Determine whether a given set with two operations is a vector space.
- Recognize standard examples of vector spaces: nth dimensional Euclidean space, the set of all
*m*x*n*matrices, the set of all polynomials, the set of all polynomials of degree ≤*n*, the set of all continuous functions defined on the real line, the set of all continuous functions defined on a closed interval [*a,b*]. - Determine whether a given subset of a vector space is a subspace .
- Recognize subspaces of R
^{2}and R^{3}and understand their geometric interpretations. - Determine whether a vector is a linear combination of a given finite set of vectors in a vector space and be able to write this linear combination.
- Determine whether a given set of vectors in a vector space is a spanning set for that vector space.
- Determine whether a given finite set of vectors in a vector space is linearly independent.
- Determine whether a given set of vectors in a vector space forms a basis for that vector space and recognize standard bases in the vector spaces nth dimensional Euclidean space, the set of all
*m*x*n*matrices, and the set of all polynomials of degree ≤*n*. - Find the dimension of a subspace.
- Find a basis for the column or row space of a matrix.
- Find a basis for and the dimension of the nullspace of a matrix.
- Find the coordinate matrix for a vector relative to a basis.
- Find the transition matrix from one basis to another basis.
- For a given vector
**v**, find its length, a unit vector in the same or opposite direction, all vectors that are orthogonal to**v**, and the projection of**v**onto a given vector or vector space. - Find the distance, the dot product, the inner product, the cross product and the angle between any two given vectors in a Euclidean space.
- Verify and use the Cauchy-Schwarz Inequality, the Triangle Inequality and the Pythagorean Theorem for vectors.
- Determine whether any two given vectors are orthogonal, parallel, or neither.
- Determine whether a given set of vectors is orthogonal, orthonormal, or neither.
- Determine whether given subspaces are are orthogonal.
- Find the orthogonal complement of a given subspace.
- Apply the Gram-Schmidt orthonormalization process to find an orthonormal basis for a given basis, subspace, or inner product space.
- Find an orthonormal basis for the solution space of a homogeneous system of linear equations.

- Perform vector operations for vectors in R
- Linear Transformations
- Find the image and preimage of a given function.
- Determine whether a function from one vector space to another is a linear transformation.
- For a given linear transformation, find the kernel and range, find the basis for the kernel and range, and determine the nullity and rank.
- Determine whether a given linear transformation is one-to-one or onto.
- Determine whether two vector spaces are isomorphic.
- Find the standard matrix for a given linear transformation and the composition of linear transformations.
- Determine whether a given linear transformation is invertible and find its inverse if it exists.
- Know and use the properties of similar matrices.

- Eigenvalues and Eigenvectors
- Verify an eigenvalue and an eigenvector of a given matrix.
- Understand the geometrical interpretation of the eigenvalue and eigenvector of a given matrix.
- Find the characteristic equation and the eigenvalues and corresponding eigenvectors of a given matrix.
- Determine whether a given matrix is diagonalizable, symmetric, or orthogonal.
- Find (if possible) a nonsingular matrix
*P*for a given matrix*A*such that*P*^{-}^{1}*AP*is diagonal. - Find a basis
*B*(if possible) for the domain of a linear transformation*T*such that the matrix of*T*relative to*B*is diagonal. - Find the eigenvalues of a given symmetric matrix and determine the dimension of the corresponding eigen space.
- Find an orthogonal matrix that diagonalizes a given matrix.

- Applications
- Applications of the above objectives to geometry and other topics which could include differential equations, systems of linear differential equations, least squares approximations, Fourier approximations, population growth or models in dynamical systems.