Index Index
Basis, Definition
change-of-basis, Paragraph
left-handed, Definition
negatively oriented, Definition
positively oriented, Definition
right-handed, Definition Paragraph
standard basis, Paragraph Definition
definition
Change of basis matrix, Definition
Characteristic Polynomial, Definition
definitions
Basis, Definition
left-handed, Definition
negatively oriented, Definition
positively oriented, Definition
right-handed, Definition
standard basis, Definition
Determinant, Definition
Dot product
algebraic definition, Paragraph
geometric definition, Paragraph
Eigenspace, Definition
Eigenvalue, Definition
algebraic multiplicity, Definition
geometric multiplicity, Definition
Eigenvector, Definition
Equivalent systems, Definition
Function
composition, Definition
identity function, Definition
inverse of, Definition
invertible function, Definition
Homogeneous system, Definition
Hyperplane, Definition
Injective function, Definition
Line
direction vector for, Definition
normal form, Definition
normal vector, Definition
vector form of, Definition
Linear combination, Definition
coefficient of, Definition
convex, Definition
non-negative, Definition
trivial, Definition
Linear dependence/independence
algebraic definition, Definition
geometric definition, Definition
Linear transformation, Definition
induced, Definition
null space (kernel), Definition
nullity, Definition
orientation preserving, Definition
orientation reversing, Definition
range (image), Definition
rank, Definition
representation in a basis, Definition
Matrix
column space, Definition
diagonal, Definition
diagonal of, Definition
diagonalizible, Definition
elementary matrix, Definition
fundamental subspaces of, Definition
identity matrix, Definition Definition
inverse of, Definition
linear transformation induced by, Definition
lower triangle of, Definition
lower triangular, Definition
null space, Definition
nullity, Definition
of a linear transformation, Definition
rank, Definition
row space, Definition
similar matrices, Definition
skew-symmetric, Definition
square, Definition
symmetric, Definition
transpose of, Definition
triangular, Definition
upper triangle of, Definition
upper triangular, Definition
zero matrix, Definition
One-to-one function, Definition
Onto function, Definition
Orthogonal, Definition
Parameter variable, Paragraph
Plane
direction vector for, Definition
vector form of, Definition
Projection, Definition
Scalar, Paragraph
Set
empty, Paragraph
equality, Definition
image of, Definition
intersection, Definition
union, Definition
Set sum, Definition
Set-builder notation, Paragraph
Span, Definition
Subset, Definition
Subspace
dimension, Definition
trivial subspace, Definition
Subspaces, Definition
Superset, Definition
Surjective function, Definition
Vector, Paragraph
representation in a basis, Definition
Vector component, Definition
Vector operations
addition, Paragraph
scalar multiplication, Paragraph
Zero vector (\(\vec 0\)), Definition
Determinant, Definition
\(2\times 2\) formula, Theorem
\(3\times 3\) formula, Theorem
Diagonalizable, Paragraph
Dimension, Paragraph
Displacement, Paragraph
Dot product, Paragraph
algebraic definition, Paragraph
geometric definition, Paragraph
Eigenspace, Definition
Eigenvalue, Definition
algebraic multiplicity, Definition
geometric multiplicity, Definition
Eigenvector, Definition Theorem
Function
composition, Definition
identity function, Definition
inverse of, Definition
invertible function, Definition
Heisenberg matrix, Exercise
Hyperplane, Definition
Line
direction vector for, Definition
normal form, Definition
normal vector, Definition
parallel, Example
skew, Example
vector form of, Definition
Linear combination, Definition
coefficient of, Definition
convex, Definition
non-negative, Definition
trivial, Definition
Linear transformation, Definition Paragraph
composition of, Paragraph
induced, Definition
notation, Paragraph
nullity, Definition
orientation preserving, Definition
orientation reversing, Definition
proving linearity, Takeaway
range (image), Definition
rank, Definition
representation in a basis, Paragraph Definition
Linear transformations
null space (kernel), Definition
Linearity, Paragraph
Linearly dependent, Definition
Linearly independent, Definition
Matrix
coefficient matrix, Paragraph
column space, Definition
definition, Paragraph
diagonal matrix, Theorem
diagonalizable, Definition
elementary matrix, Definition Paragraph Paragraph
fundmamental subspaces of, Definition
identity matrix, Definition Definition
inverse of, Definition
linear transformation induced by, Definition
matrix multiplication, Paragraph
matrix transformation, Paragraph
null space, Definition
nullity, Definition
of a linear transformation, Definition
rank, Definition
reduced row echelon form, Section
row space, Definition
similar matrices, Paragraph
transpose of, Definition
Matrix equations
the column picture, Paragraph
the row picture, Paragraph
One-to-one function, Definition
Onto function, Definition
Orthogonal, Definition
Outer product, Item
Parameter variable, Paragraph
Plane
direction vector for, Definition
normal vector, Definition Definition
scalar form of, Paragraph
vector form of, Definition
Point
in Euclidean Space, Paragraph
of a set, Paragraph
Projection, Definition
Scalar, Paragraph
Scalar product, Paragraph
Set
element of, Paragraph
empty, Paragraph
equality, Definition
image of, Definition
intersection, Definition
union, Definition
Set sum, Definition
Set-builder notation, Paragraph
Span, Paragraph Definition
translated, Paragraph
Standard basis, Paragraph
Subspace, Definition Theorem
dimension, Definition
trivial subspace, Definition
Surjective function, Definition
symbols
\([\mathcal{T}]_{\mathcal{B}}\), Definition
\([\vec v]_{\mathcal{B}}\), Definition
\(\Abs{\mathcal{T}}\), Definition
\(\BasisChange{\mathcal{A}}{\mathcal{B}}\), Definition
\(\cap\), Definition
\(\chr(A)\), Definition
\(\Col(M)\), Definition
\(\cup\), Definition
\(\det(\mathcal{T})\), Definition
\(\emptyset\), Paragraph
\(\Ident\), Definition
\(\in\), Paragraph
\(\N\), Display Mathematics
\(\norm{\:\cdot\:}\), Paragraph
\(\notin\), Paragraph
\(\Null(M)\), Definition
\(\Nullity(M)\), Definition
\(\Proj_{X}\vec v\), Definition
\(\Q\), Display Mathematics
\(\R\), Display Mathematics
\(\R^{n}\), Display Mathematics
\(\Rank(M)\), Definition
\(\Rank(T)\), Definition
\(\Row(M)\), Definition
\(\sim\), Definition
\(\Span V\), Definition
\(\subseteq\), Definition
\(\Vcomp_{\vec v}\vec u\), Definition
\(\vec 0\), Definition
\(\vec a\cdot \vec b\), Paragraph
\(\Z\), Display Mathematics
\(C_{n}\), Definition
\(g\circ f\), Definition
\(L(X)\) (Image of a set), Definition
\(M^{T}\), Definition
System of linear equations
coefficient matrix, Paragraph
equivalent, Example
homogeneous, Definition
overdetermined system, Example
True vector, Paragraph
Vector, Paragraph
direction of, Paragraph Definition
norm of, Paragraph
positive direction of, Definition
representation in a basis, Definition
rooted, Paragraph
Vector component, Definition
Vector operations
addition, Paragraph
laws of, Paragraph
scalar multiplication, Paragraph
theorem II, Theorem
Zero vector (\(\vec 0\)), Definition Theorem
