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# 22A Summer II 2009 Final Review Sheet

Be able to state (with words and symbols) the definition
of the following terms

• Subspace

• linear combination of the vectors .

• A set of vectors span a vector space

• Basis, dimension

• The nullspace, column space, row space for an m × n matrix A

• The rank and nullity of a matrix A

• eigenvalues for a matrix A

• eigenvectors associated to an eigenvalue λ for a matrix A

• The associated eigenspace for an eigenvalue λ for a matrix A

• characteristic polynomial of a matrix A

• symmetric, diagonal, upper-triangular, lower-triangular, and orthogonal
matrices

• diagonalizability of a matrix A and orthogonally diagonalizabity of A

• Linear transformation from Rm to Rn

• The standard matrix for a linear transformation

• range of a linear transformation

• orthogonal vectors in Rn, an orthonormal set of vectors in Rn and
orthonormal basis in Rn

• a 1-1 function

• a onto function

• The coordinate vector with respect to a given ordered basis

• The transition matrix ( or change of coordinate matrix) from one ordered
basis to another ordered basis

Know the statement of and how to use

• Rank Nullity Theorem

• Algebraic properties of matrix operations

• Algebraic properties of inverse matrices

• Algebraic properties of determinants

• Algebraic properties of vector operations

• Algebraic properties of linear transformations

• Algebraic properties of abstract vector space operations

• Algebraic properites of eigenvectors

• The various equivalences for nonsingularity for a matrix A (see pg. 204
in the book)

• The algebraic properties of coordinate vectors and transition matrices

Be able to

• Prove or disprove a statement about abstract operations

• Be able to determine if a subset of a vector space V is a subspace of V

• Be able to determine if a set of vectors   in a vector space
V

– spans V

– is linearly independent

– is a basis for V

• Solve for the general solution for a linear system

• Recognize row echelon forms and reduced row echelon forms (RREF)

• Use elementary row operations to get a RREF

• Understand what a REF tells us about the solutions to a linear system

• Be able to compute the general solution for a homogeneous system

• Be able to compute determinants

• Be able to determine whether or not a given matrix is invertible and if
so find its inverse

• Be able to find a basis for the null space of A

• Be able to find a basis for the column space and row space of A

• Be able to compute the rank and nullity of a matrix A

• Be able to find a basis for W = span{}

• Be able to use Gram-Schmidt to orthonormalize a basis

• Be able to verify if a set of vectors is orthonormal and know examples
of orthonormal basis

• Be able to compute eigenvalues of a matrix A

• Be able to find associated eigenvectors for a given eigenvalue

• Be able to find a basis for the eigenspace Eλ for a matrix A

• Be able to determine if a matrix A is diagonalizable and if so be able
to find a matrix P so that P-1AP is diagonal

• For a symmetric matrix A be able to find orthogonal matrix P so that
P-1AP is diagonal

• Be able to verify if matrix A is orthogonal

• Be able to determine if a vector is a linear combination of other vectors

• Be able to determine if another vector is in the range of a linear transformation

• Be able to find a basis for the range of a linear transformation

• Be able to find the standard matrix of a linear transformation

• Be able to recognize whether or not a given function is linear or not
linear

• Determine if a linear transformation is 1-1 and/or onto.

• Understand the various equivalences for nonsingularity for a matrix A
(see pg. 204 in the book)

• Understand the various algebraic properties of the various operations
in linear algebra and how to give basic algebraic proofs using these
properties

• Understand how to setup linear system in order to answer various linear
algebra questions

• Be able to find the coordinate vector with respect to a given ordered
basis

• Be able to find the transition matrix from one ordered basis to another
ordered basis

• Be able to find the vector in a vector space that a given coordinate
vector with respect to an ordered basis represents

Be sure to review

• All lecture notes

• All WebWork assignments

• All reading assignments (except Sec. 1.8)

• All suggested problems

DO NOT

• Wait until the night before to study

• Merely memorize problems. Try to understand the methods and concepts
involved.

• Only Study the old Final and expect to succeed.