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Dec 08, 2024
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MAT 148 - Linear Algebra w/Applications Credits: 4 Lecture Hours: 4 Lab Hours: 0 Practicum Hours: 0 Work Experience: 0 Course Type: General A study of the use and application of matrices in the solution of systems of linear equations, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, bases and projections. Linear algebra is a core course in many engineering, physics, mathematics and computer science programs. This course makes heavy use of computing technology. Graphing calculators required. Prerequisite: MAT 211 with a C- or better Competencies
- Solve a system of linear equations
- Identify whether a system of linear equations has none, one or infinitely many solutions
- Solve a system of linear equations using row-reduction and back-substitution
- Perform arithmetic operations with vectors
- Discuss the span of a set of vectors
- Solve the matrix equation Ax=b
- Decide whether a set of vectors is linearly independent
- Discuss linear transformations from one Euclidean space to another
- Identify whether a transformation is one-to-one
- Perform operations on matrices
- Compute a linear combination of matrices
- Computer the product of two matrices, if it exists
- Write the transpose of a matrix
- Find the inverse of a matrix, if it exists
- Perform arithmetic operations on partitioned matrices
- Discuss various Euclidean spaces
- Define what a Euclidean space is
- Find the dimension of a given space
- Define what a subspace is and state conditions for its existence
- Find the column space of a matrix
- Find the row space of a matrix
- Find the nullspace of a matrix
- State the rank of a matrix and relate it to the column and row spaces of the matrix
- Discuss the determinant of a matrix
- Compute the determinant of a matrix by using a cofactor expansion across a row or down a column
- Describe how various row operations on a matrix affect its determinant
- Use Cramer’s rule to solve a system of linear equations
- Use the determinant of a matrix to determine whether a matrix is invertible
- Discuss vector spaces
- State the definition of a vector space
- State the definition of a vector subspace
- Define what is meant by a linear transformation from one vector space to another
- Decide whether a given set of vectors is linearly independent
- Find the coordinate vector in one coordinate system with respect to another coordinate system
- Find a set of vectors which form a basis of a vector space
- State the dimension of a given vector space
- Use the Rank Theorem to relate the rank of a matrix and the dimension of its nullspace
- State several conditions equivalent to a matrix being invertible
- Find the change-of-coordinate matrix from one coordinate system to another
- Discuss the eigenspace of a matrix
- Find the characteristic polynomial of a matrix
- find the eigenvalues, real and complex, of a matrix
- For each eigenvalue of a matrix, find a corresponding eigenvector
- use eigenvalues and eigenvectors to diagonalize a matrix
- Write the Jordan form of a matrix
- Perform vector operations in n-dimensional space
- Compute the inner product of two column matrices or, equivalently, the dot product of two vectors
- Calculate the angle between two vectors
- Compute the length (or norm) of a vector
- Decide whether a given set of vectors forms an orthonormal basis of a vector space
- Find the orthogonal projection of one vector onto another
- given a set of linearly independent vectors spanning a vector space, use the Gram-Schmidt process to find an orthogonal basis for that space
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