MAT 242 Introductory Linear Algebra

Assumes use of a scientific calculator.

1. Add/subtract/multiply matrices if possible. Calculate determinants of square matrices including size 4x4.
2. 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.
3. Factor a 2x2 matrix to a product of elementary matrices.
4. Define a vector space, and determine if a set forms a vector space or not.
5. Determine whether subsets of vector spaces are subspaces or not.
6. Determine whether a set of vectors (matrices, polynomials, etc…) in a vector space is linearly independent or linearly dependent.
7. 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.
8. Define the terms “basis” and “dimension” of a vector space.
9. Find a basis for the row space, column space and solution space of a matrix.
10. Find the transition matrix from one basis of a vector space to another basis.
11. Given vectors in, be able to add, subtract, find the dot product, find the angle between these and find the distance between these.
12. Define an inner product space and justify whether a set forms an inner product space or not.
13. 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.
14. Define a linear transformation and justify whether a transformation is linear or not.
15. Find a basis for the kernel and range of a linear transformation.
16. Find the matrix of a linear transformation.
17. Justify whether a linear transformation is invertible or not, and if so, find the rule for the inverse transformation.
18. Find the eigenvalues and eigenvectors of a 3x3 matrix.
19. Define diagonalizable matrix and determine whether a square matrix A is diagonalizable or not. If so, then find a matrix which diagonalizes A.
20. Find an orthogonal matrix which diagonalizes a symmetric matrix.

Objectives

1. Perform operations on matrices when appropriate, including inversion, determinants, factoring to elementary products and finding the basis for the row and column space.
2. Define a vector space and inner product space and determine if sets of vectors form a vector space, inner product space or subspace.
3. Determine whether vectors are linearly independent, orthogonal, and orthonormal and use the Gram-Schmidt process to orthonormalize when appropriate.
4. Determine the eigenvalues and eigenvectors of a matrix when appropriate and find an orthogonal matrix that diagonalized a symmetric matrix.
5. Determine if transformations are linear, find the matrix of a linear transformation and find a basis for the kernel and range.