Algebra Final Exam Study Guide

Chapter 1 - Vectors

Draw vectors in R2 and R3 as well as graphically represent a sum, difference, or scalar
multiple of a vector .
Compute sums , differences , scalar multiples, length, and dot products of vectors algebraically.
Two definitions of dot product and be able to find the angle between two vectors.
Properties of the dot product (Thm 1.2) and length (Thm 1.3).
• Definition of orthogonal, both geometrically and in terms of dot product.
• Compute the projection of a vector onto another vector and be able to represent the
projection geometrically.
Equations of lines (vector and parametric) and planes (vector and general).

Chapter 2 - Systems of Linear Equations

• Use Gauss- Jordan elimination (both by hand and with a calculator) to solve a system
of linear equations .
• Find the span of a set of vectors.
• Determine if a set of vectors is linearly independent.

Chapter 3 - Matrices

• Find the sum, difference , or product of two matrices . Find the scalar multiple, power
or transpose of a matrix.
• Definition of symmetric and skew-symmetric.
• Properties of matrix addition and scalar multiplication (Thm 3.2), matrix multiplication
(Thm 3.3), and the transpose (Thm 3.4).
• Definition of the inverse of a matrix.
• Properties of the inverse (Thm 3.9)
• Fundamental Theorem of Invertible Matrices (Thm 3.12, 3.27, 4.17)
• Find the inverse of a 2 × 2 matrix using special formula or the inverse of an n × n
matrix using the Gauss- Jordan method .
• Definition of subspace of Rn, determine if a set is a subspace of Rn.
• Definition of basis, dimension, rank and nullity. For a set of vectors, find a basis and
its dimension. Find the rank and nullity of a matrix.
• The Rank Theorem (Them 3.26).
• Definition of linear transformation (from Rn to Rm) and matrix transformation.
• Every linear transformation T : Rn -> Rm is a matrix transformation and vice versa.
(Thm 3.30, 3.31) Find the standard matrix of a linear transformation.
• Find the composition and inverse of a linear transformation using the standard matrix
of the transformation.

Chapter 4 - Eigenvalues and Eigenvectors

• Definition of eigenvalue, eigenvector, eigenspace.
• Compute the determinant of and n × n matrix.
• Properties of the determinant (Thm 4.3,4.7,4.8,4.9,4.10).
• Find eigenvalues, eigenvectors, and eigenspaces of an n × n matrix.
• Definition of algebraic and geometric multiplicity of an eigenvalue.
• Definition of similarity, properties of similarity (Thm 4.21, 4.22)
• Definition of diagonalizable.
• Determine if a matrix is diagonalizable and if it is, find an invertible matrix P and a
diagonal matrix D so that P−1AP = D.

Chapter 5 - Orthogonality

• Definition of orthogonal set, orthogonal basis, orthonormal set and orthonormal basis.
• Definition of orthogonal matrix, properties of orthogonal matrices (Thm 5.6,5.7,5.8).
• Definition of orthogonal complement.
• Compute the orthogonal complement of a subspace W of Rn.

Chapter 6 - Vector Spaces

• Definition of vector space (10 axioms). Determine if a set with two given operations is
a vector
• Properties of vector spaces (Thm 6.1)
• Definition of subspace. Determine if a set is a subspace of a given vector space.
• Find the span of a set of vectors.
• Determine if a set of vectors is linearly independent.
• Definition of basis and dimension of a vector space.
• Properties of basis/span/L-I of vector space (Thm 6.10).
• Definition of linear transformation on a vector space.
• Properties of linear transformation (Thm 6.14).
• Definition of composition and inverses of linear transformations.
• Definition of kernel and range of a linear transformation. Be able to find the kernel
and range of a linear transformation.
• Definition of rank and nullity of a linear transformation. Be able to find the rank and
nullity of a linear transformation.
• The Rank Theorem (Thm 6.19).
• Definition of one-to-one and onto linear transformation. Determine if a linear transformation
is one-to-one and/or onto.
• Definition of isomorphism. Determine if a transformation is an isomorphism. Determine
if two vector spaces are isomorphic.

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