# Algebra Final Exam Study Guide

## Chapter 1 - Vectors

• Draw vectors in R^{2} and R^{3} 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 R^{n}, determine if a set is a subspace of R^{n}.

• 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 R^{n} to R^{m}) and
matrix transformation.

• Every linear transformation T : R^{n} -> R^{m} 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^{−1}AP = 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 R^{n}.

## Chapter 6 - Vector Spaces

• Definition of vector space (10 axioms). Determine if a
set with two given operations is

a vector space.

• 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.

Prev | Next |