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Syllabus for Linear Algebra


An applications approach to introductory linear algebra. Covers systems of linear equations,
matrices, linear independence, vector spaces, inner product spaces, linear transformations,
eigenvalues, eigenvectors and applications.

Semester Offered: On Demand
Prerequisites: Grade of “C” or better in Math 189.

Common Student Learning Outcomes
Upon successful completion of San Juan College programs and degrees, the student will....


Students will actively and independently acquire, apply and adapt skills and
knowledge to develop expertise and a broader understanding of the world as lifelong


Students will think analytically and creatively to explore ideas, make connections,
draw conclusions, and solve problems.


Students will exchange ideas and information with clarity and originality in multiple


Students will demonstrate proficiency in the use of technologies in the broadest sense
related to their field of study.


Students will act purposefully, reflectively, and respectfully in diverse and complex

Upon completion of the course, the student should have a working knowledge of the following:

1.) Systems of Linear Equations & Applications
2.) Matrices & Determinants & Applications
3.) Vector Spaces & Applications
4.) Inner Product Spaces & Applications
5.) Linear Transformations & Applications
6.) Eigenvalues & Eigenvectors & Applications

Upon completion of this course, the student should be able to:

1.1 Solve systems of linear equations by elimination & substitution.
1.2 Solve systems of linear equations using Gaussian & Gauss-Jordan elimination.
1.3 Apply the theory of systems to a variety of problems.
2.1 Perform matrix addition, subtraction and multiplication.
2.2 Find the inverse, transpose, determinant, adjoint and trace of a matrix.
2.3 Utilize matrix algebra to solve systems of equations.
2.4 Use matrices in applications

3.1 Verify that a set has the structure of a vector space.
3.2 Discuss and verify the linear independence/dependence of a set of vectors.
3.3 Understand and work with bases of Euclidean as well as abstract vector spaces.
3.4 Change Basis Coordinates using transition matrices.
3.5 Use Change of Basis in Applications.

4.1 Verify that a functional is an inner-product.
4.2 Understand the importance of orthonormal bases.
4.3 Construct orthonormal bases using the Gram-Schmidt algorithm.
4.4 Find coordinates in orthonormal bases.
4.5 Construct the metric matrix for a given basis in an abstract vector space
and use it to compute inner products.
4.6 Use Inner Product Spaces in applications.

5.1 Verify that a function between vector spaces is a linear transformation.
5.2 Find the kernel and range of a linear transformation.
5.3 Calculate the matrix of a linear transformation with respect to a given basis.
5.4 Use the matrix of a linear transformation to analyze it.
5.5 Use transition matrices to find the matrix of a linear transformation.
5.6 Use linear transformations in applications.

6.1 Understand the importance of eigenvalues.
6.2 Find the eigenvalues and eigenvectors of a matrix
6.3 Use eigenvalues to diagonalize matrices.
6.4 Deal with symmetric matrices and orthogonal diagonalization.
6.5 Use eigenvalues to solve problems in differential equations, stochastic processes,
quadratic forms and transcendental functions of matrices.

7.1 Write and construct simple mathematical proofs.


The TI-82, TI-83, TI-84, TI-85 or TI-86 graphing calculator is required for the course. A TI-83 Plus or
TI-84 Plus Graphing Calculator
is strongly recommended. Graphing calculators capable of symbolic
manipulation (such as TI-89 or TI-92 and other such calculators) will not be allowed on examinations, the
final exam and where the instructor finds fit.