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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
• 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
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
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 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.