## Vocabulary

A list of important linear algebra terms, listed by chapter and section of your textbook. Please make sure you understand the meaning of each of these terms. They form the building blocks for the conceptual framework of the subject.

**Chapter 1****Section 1.1**- linear equation
- coefficient
- system of linear equations, linear system
- solution, solution set
- equivalent (linear systems)
- consistent, inconsistent
- coefficient matrix
- augmented matrix
- elementary row operations
- row equivalent (matrices)

**Section 1.2**- leading entry
- echelon form, reduced echelon form
- row reduced, row reduction
- pivot position, pivot column
- basic variable
- free variable

**Section 1.3**- vector
- scalar, scalar multiple
- zero vector
- linear combination
- Span{
**v**_{1}, ... ,**v**_{p}}, subset of**R**^{n}spanned by**v**_{1}, ... ,**v**_{p}

**Section 1.4***A***x**(for*A*an*m*x*n*matrix and**x**in**R**^{n})- matrix equation
*A***x**=**b** - spans
- identity matrix

**Section 1.5**- homogeneous
- trivial solution, nontrivial solution (of a homogeneous linear system)
- parametric vector form

**Section 1.7**- linearly independent
- linearly dependent, linear dependence relation

**Section 1.8**- transformation, function, mapping
- domain
- codomain
- range
- linear transformation

**Section 1.9**- standard matrix for a linear transformation
- onto
- one-to-one

**Chapter 2****Section 2.1**- diagonal entries, main diagonal
- diagonal matrix
- zero matrix
- equal (matrices)
- sum (of matrices)
- scalar multiple (of a matrix)
- matrix multiplication, row-column rule
- associative law
- left distributive law, right distributive law
- identity (for matrix multiplication), identity matrix
- commute
- transpose

**Section 2.2**- invertible, nonsingular
- inverse
- singular
- determinant
- elementary matrix
- algorithm for finding
*A*^{-1}

**Section 2.3**- the invertible matrix theorem
- invertible linear transformation

**Chapter 4****Section 4.1**- vector space
- subspace
- zero subspace
- linear combination
- Span{
**v**_{1}, ...,**v**_{p}}, subspace spanned (or generated) by {**v**_{1}, ...,**v**_{p}}. - spanning (or generating) set for a subspace

**Section 4.2**- Nul(
*A*), null space of a matrix*A* - Col(
*A*), column space of a matrix*A* - kernel of a linear transformation
- range of a linear transformation

- Nul(
**Section 4.3**- linearly independent, linearly dependent, linear dependence relation
- basis
- standard basis for
**R**^{n} - standard basis for
**P**_{n} - spanning set theorem

**Section 4.5**- finite-dimensional vector space, dimension
- infinite-dimensional vector space
- the basis theorem

**Section 4.6**- row space of a matrix
- rank of a matrix
- nullity of a matrix
- the rank theorem
- the invertible matrix theorem (continued)

**Chapter 3****Section 3.1**- determinant
*(i,j)*-cofactor- cofactor expansion

**Section 3.2**- determinant and row operations
- invertible if and only if nonzero determinant
- determinant of transpose
- multiplicative property

**Chapter 5****Section 5.1**- eigenvector
- eigenvalue
- eigenvector corresponding to λ
- eigenspace of
*A*corresponding to λ

**Section 5.2**- characteristic equation
- characteristic polynomial
- multiplicity of an eigenvalue
- similar matrices

**Section 5.3**- diagonalizable
- eigenvector basis

**Chapter 6****Section 6.1**- inner product, dot product
- length, norm
- distance
- orthogonal
- Pythagorean theorem
- orthogonal complement