MAT 242 Introductory Linear Algebra
Assumes use of a scientific calculator.
- Add/subtract/multiply matrices if possible. Calculate determinants of square matrices including size 4x4.
- Solve linear systems by Gaussian Elimination, Cramer’s Rule or the inverse matrix method, if possible. If the system has infinitely many solutions, find the general solution.
- Factor a 2x2 matrix to a product of elementary matrices.
- Define a vector space, and determine if a set forms a vector space or not.
- Determine whether subsets of vector spaces are subspaces or not.
- Determine whether a set of vectors (matrices, polynomials, etc…) in a vector space is linearly independent or linearly dependent.
- Show whether or not a given vector is in the span of a set of vectors, and if so, write the given vector as a linear combination of the vectors in the set.
- Define the terms “basis” and “dimension” of a vector space.
- Find a basis for the row space, column space and solution space of a matrix.
- Find the transition matrix from one basis of a vector space to another basis.
- Given vectors in, be able to add, subtract, find the dot product, find the angle between these and find the distance between these.
- Define an inner product space and justify whether a set forms an inner product space or not.
- Show whether a set of vectors is orthogonal, orthonormal or neither. Use the Gram-Schmidt orthonormalization process to transform a set of vectors into an orthonormal set.
- Define a linear transformation and justify whether a transformation is linear or not.
- Find a basis for the kernel and range of a linear transformation.
- Find the matrix of a linear transformation.
- Justify whether a linear transformation is invertible or not, and if so, find the rule for the inverse transformation.
- Find the eigenvalues and eigenvectors of a 3x3 matrix.
- Define diagonalizable matrix and determine whether a square matrix A is diagonalizable or not. If so, then find a matrix which diagonalizes A.
- Find an orthogonal matrix which diagonalizes a symmetric matrix.
- 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.