MAT 252 - Introduction to Linear Algebra

1. Perform operations with matrices, calculate determinants, find eigenvalues and eigenvectors, and use matrices to solve systems of linear equations.
2. Define vector spaces and find a basis for a subspace.
3. Determine the matrix of a linear transformation with respect to a given basis, its kernel and range, and perform operations with linear transformations.

1. Use matrices to solve systems of linear equations; perform operations with matrices, calculate the inverse of a non-singular matrix, and calculate the determinant of a square matrix.
2. Define a vector space and perform vector operations; determine linear independence and find a spanning set of vectors.
3. Define subspaces of a vector space; find a basis for a subspace and determine its dimension; find the subspaces associated with a matrix, and determine the rank and nullity of a matrix.
4. Define a linear transformation and find the matrix associated with it; determine the kernel and range of a transformation; find the inverse of a transformation and the composition of two or more linear transformations; calculate the change of basis matrix.
5. Find the eigenvalues and eigenvectors of a matrix; determine similarity between two matrices; diagonalize a matrix.
6. Use the Gram-Schmidt process to obtain an orthogonal and an orthonormal basis; define an inner product space.
7. Use Linear Algebra in various scientific and mathematical applications.

1. Matrices and Systems of Linear Equations
   1. Gaussian and Gauss-Jordan elimination
   2. Matrix operations
   3. Inverse and determinant of square matrices
   4. Applications
2. Vector Spaces
   1. Definition
   2. Algebra of vectors
   3. Linear independence
   4. Spanning sets of vectors
3. Subspaces
   1. Definition
   2. Basis and dimension
   3. Subspaces associated with a matrix
   4. Rank and nullity of a matrix
4. Linear Transformations
   1. Definition
   2. Kernel and range
   3. Matrix of a linear transformation
   4. Composition and inverses of linear transformations
   5. Change of Basis
   6. Applications
5. Eigenvalues and Eigenvectors
   1. Definition
   2. Similar matrices
   3. Diagonalization of matrices
6. Orthogonality and Inner Product Spaces
   1. Orthogonal and orthonormal basis
   2. Orthogonal projections
   3. Gram-Schmidt process
   4. Orthogonal diagonalization of symmetric matrices
   5. Applications
   6. Definition of inner product spaces  
7. Applications
   1. Matrices and systems of linear equations
   2. Linear transformations
   3. Orthogonality and inner product spaces