# MAT 242 Introductory Linear Algebra

Exit Skills

Assumes use of a scientific calculator.

1. Add, subtract, and multiply matrices if possible. Calculate determinants by cofactor expansion and row reduction and use appropriate properties.
2. 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.
3. Factor a square matrix to a product of elementary matrices.
4. Define a vector space and determine if a set forms a vector space.
5. Determine whether subsets of vector spaces are subspaces.
6. Determine whether a set of vectors in a vector space is linearly independent or linearly dependent.
7. 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.
8. 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.
9. Find the transition matrix from one basis of a vector space to another basis.
10. Find real and complex eigenvalues and eigenvectors of a square matrix.  Find bases for eigenspace.
11. Define diagonalizable matrix and determine whether a square matrix A is diagonalizable. If so, find a matrix which diagonalizes A.
12. Define an inner product space and determine whether a set of vectors forms an inner product space.
13. 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.
14. Find an orthogonal matrix which diagonalizes a symmetric matrix.
15. Define and recognize Hermitian, Unitary, and Normal matrices.
16. Define a linear transformation and determine whether a transformation is linear.
17. Find a basis for the kernel and range of a linear transformation.
18. Determine whether a linear transformation is invertible.  If so, find the rule for the inverse transformation.
19. Find the matrix of a linear transformation.

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.

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