Math 18 Vocabulary

Updated 4/14/21

A list of important linear algebra terms taken from Linear Algebra and Its Applications (fifth edition), by David C. Lay, Steven R. Lay, and Judi J. McDonald; published by Pearson 2014 (and organized by Chapter and Section of that book). It is highly recommended that you review these terms to make sure you're familiar with them.

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• 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
    • 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
    • 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

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