This is a list of important linear algebra terms, listed by chapter and section of your textbook. It is imperative that you learn the meaning of each of these terms. They form the building blocks of the conceptual framework for the subject.

• One way to learn the vocabulary is to make flash cards and review them by yourself or with fellow students in the course.
• If you review your flash cards at a Starbucks and someone asks what foreign language you're studying, you can say "linear algebra."

• 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{v₁, ... , v_p}, subset of Rⁿ spanned by v₁, ... , v_p
  • Section 1.4
    • Ax (for A an m x n matrix and x in Rⁿ)
    • matrix equation Ax = b
    • spans
    • identity matrix
  • Section 1.5
    • homogeneous
    • trivial solution, nontrivial solution (of a homogeneous linear system)
    • parametric vector equation, parametric vector form
  • Section 1.7
    • linearly independent
    • linearly dependent, linear dependence relation
  • Section 1.8
    • transformation, function, mapping
    • domain
    • codomain
    • image, range
    • linear, linear transformation
    • shear transformation
    • dilation
    • rotation
  • 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{v₁, ..., v_p}, subspace spanned (or generated) by {v₁, ..., 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
  • Section 4.3
    • linearly independent, linearly dependent, linear dependence relation
    • basis
    • standard basis for Rⁿ
    • standard basis for P_n
    • spanning set theorem
  • Section 4.4
    • the unique representation theorem
    • coordinates of a vector x relative to a basis B
    • coordinate vector of x (relative to B)
    • coordinate mapping
    • change-of-coordinates matrix
    • isomorphism
  • 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)
  • Section 4.7
    • change-of-coordinates matrix from B to C
• 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
  • Section 3.3
    • determinant and area
    • determinant and volume
• 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, similarity transformation
  • Section 5.3
    • diagonalizable
    • eigenvector basis
• Chapter 6
  • Section 6.1 (and 6.7)
    • inner product, dot product
    • length, norm
    • distance
    • orthogonal
    • Pythagorean theorem
    • orthogonal complement
  • Section 6.2
    • orthogonal set
    • orthogonal basis
    • orthogonal projection of y onto u, orthogonal projection of y onto L
    • component of y orthogonal to u
    • orthonormal set
    • orthonormal basis
    • orthogonal matrix
  • Section 6.3
    • orthogonal decomposition theorem
    • orthogonal projection of y onto W
    • best approximation theorem
    • best approximation of y by elements of W
  • Section 6.4
    • Gram-Schmidt process
    • QR factorization
• Chapter 7
  • Section 7.1
    • symmetric matrix
    • spectral theorem for symmetric matrices
    • spectral decomposition
    • projection matrix