MAT 148 - Linear Algebra w/Applications

1. Solve a system of linear equations
   1. Identify whether a system of linear equations has none, one or infinitely many solutions
   2. Solve a system of linear equations using row-reduction and back-substitution
   3. Perform arithmetic operations with vectors
   4. Discuss the span of a set of vectors
   5. Solve the matrix equation Ax=b
   6. Decide whether a set of vectors is linearly independent
   7. Discuss linear transformations from one Euclidean space to another
   8. Identify whether a transformation is one-to-one
2. Perform operations on matrices
   1. Compute a linear combination of matrices
   2. Computer the product of two matrices, if it exists
   3. Write the transpose of a matrix
   4. Find the inverse of a matrix, if it exists
   5. Perform arithmetic operations on partitioned matrices
3. Discuss various Euclidean spaces
   1. Define what a Euclidean space is
   2. Find the dimension of a given space
   3. Define what a subspace is and state conditions for its existence
   4. Find the column space of a matrix
   5. Find the row space of a matrix
   6. Find the nullspace of a matrix
   7. State the rank of a matrix and relate it to the column and row spaces of the matrix
4. Discuss the determinant of a matrix
   1. Compute the determinant of a matrix by using a cofactor expansion across a row or down a column
   2. Describe how various row operations on a matrix affect its determinant
   3. Use Cramer’s rule to solve a system of linear equations
   4. Use the determinant of a matrix to determine whether a matrix is invertible
5. Discuss vector spaces
   1. State the definition of a vector space
   2. State the definition of a vector subspace
   3. Define what is meant by a linear transformation from one vector space to another
   4. Decide whether a given set of vectors is linearly independent
   5. Find the coordinate vector in one coordinate system with respect to another coordinate system
   6. Find a set of vectors which form a basis of a vector space
   7. State the dimension of a given vector space
   8. Use the Rank Theorem to relate the rank of a matrix and the dimension of its nullspace
   9. State several conditions equivalent to a matrix being invertible
   10. Find the change-of-coordinate matrix from one coordinate system to another
6. Discuss the eigenspace of a matrix
   1. Find the characteristic polynomial of a matrix
   2. find the eigenvalues, real and complex, of a matrix
   3. For each eigenvalue of a matrix, find a corresponding eigenvector
   4. use eigenvalues and eigenvectors to diagonalize a matrix
   5. Write the Jordan form of a matrix
7. Perform vector operations in n-dimensional space
   1. Compute the inner product of two column matrices or, equivalently, the dot product of two vectors
   2. Calculate the angle between two vectors
   3. Compute the length (or norm) of a vector
   4. Decide whether a given set of vectors forms an orthonormal basis of a vector space
   5. Find the orthogonal projection of one vector onto another
   6. given a set of linearly independent vectors spanning a vector space, use the Gram-Schmidt process to find an orthogonal basis for that space