MAT 242 Introductory Linear Algebra
Assumes use of a scientific calculator.
- Add, subtract, and multiply matrices if possible. Calculate determinants by cofactor expansion and row reduction and use appropriate properties.
- Solve linear systems by Gaussian Elimination, Cramer’s Rule, and the inverse matrix method, if possible. If the system has infinitely many solutions, find the general solution.
- Factor a square matrix to a product of elementary matrices.
- Define a vector space and determine if a set forms a vector space.
- Determine whether subsets of vector spaces are subspaces.
- Determine whether a set of vectors in a vector space is linearly independent or linearly dependent.
- Determine whether a given vector is in the span of a set of vectors. If so, write the given vector as a linear combination of the vectors in the set.
- Define basis and dimension of a vector space. Find a basis (and dimension) for a vector space, including the row space, column space and null space of a matrix.
- Find the transition matrix from one basis of a vector space to another basis.
- Find real and complex eigenvalues and eigenvectors of a square matrix. Find bases for eigenspace.
- Define diagonalizable matrix and determine whether a square matrix A is diagonalizable. If so, find a matrix which diagonalizes A.
- Define an inner product space and determine whether a set of vectors forms an inner product space.
- Show whether a set of vectors is orthogonal, orthonormal or neither. Use the Gram-Schmidt orthonormalization process to transform a basis into an orthonormal basis.
- Find an orthogonal matrix which diagonalizes a symmetric matrix.
- Define and recognize Hermitian, Unitary, and Normal matrices.
- Define a linear transformation and determine whether a transformation is linear.
- Find a basis for the kernel and range of a linear transformation.
- Determine whether a linear transformation is invertible. If so, find the rule for the inverse transformation.
- Find the matrix of a linear transformation.
- Perform operations on matrices when appropriate, including inversion, determinants, factoring to elementary products and finding the basis for the row and column space.
- Define a vector space and inner product space and determine if sets of vectors form a vector space, inner product space or subspace.
- Determine whether vectors are linearly independent, orthogonal, and orthonormal and use the Gram-Schmidt process to orthonormalize when appropriate.
- Determine the eigenvalues and eigenvectors of a matrix when appropriate and find an orthogonal matrix that diagonalized a symmetric matrix.
- Determine if transformations are linear, find the matrix of a linear transformation and find a basis for the kernel and range.