MATH 2221 - Introduction to Linear Algebra
3 Credits
Provides an introduction to linear algebra topics including: systems of linear equations, matrices, determinants, vector spaces, linear transformations, eigenvalues and eigenvectors, and selected applications. Familiarity with a computer algebra system is expected. Use of technology will be embedded throughout the course.
Pre-Requisites MATH 1134
Major Content Areas
Linear systems, linear independence and linear transformations of real vector spaces 30% Matrix theory, determinants, general vector spaces 40% Eigenvectors, eigenvalues, orthogonality 30%
Learning Outcomes
Construct bases for the row, column and null space of a matrix. Relate their dimensions to one another, and to the rank and nullity of the matrix. Determine whether a vector is in the span of a finite collection of vectors. Create and recognize row equivalent matrices and equal matrices. Demonstrate the understanding of simple proof techniques. Solve systems of linear equations using matrix methods including Gaussian Elimination, Gauss-Jordan Elimination, and by matrix equation representation. Write LU and elementary matrix factorizations of square matrices where defined. Perform operations on matrices including addition, subtraction, multiplication, transposition, and inversion. Compute, explain, and apply key properties and definitions related to eigenvalues and eigenvectors of a matrix. Evaluate inner products, construct and identify orthogonal sets of vectors and orthogonal matrices, and illustrate the Gram-Schmidt process. Compute the coordinate vector of a vector relative to a finite basis. Express the solution to Ax = b as a translation of the null space of A when Ax = b is consistent. Interpret the determinant of a matrix and its properties, and apply them to linear independence, areas, volumes, orientation, invertibility, Cramer’s Rule, and the adjoint of a matrix. Create a basis for a nonzero finite dimensional vector space and find its dimension. Construct matrix representations for linear transformations relative to various bases when the domain and codomain are finite dimensional over the same field, and create change of basis matrices. Identify symmetric, skew-symmetric, lower triangular, upper triangular, triangular, scalar, and diagonal matrices and apply their basic properties. Identify a vector space from the axioms, and prove that a non-empty subset of a vector space is a subspace. Prove or disprove that a given finite set of vectors is linearly independent.
Minnesota Transfer Curriculum (MNTC) Goals
02 - Critical Thinking 04 - Mathematical/Logical Reasoning
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