{
 "source": "https://peabody.kids/flashcards/math-linear-algebra",
 "id": "math-linear-algebra",
 "title": "Linear Algebra Terms",
 "category": "math",
 "subject": "Math",
 "levels": [
  "college",
  "9-12"
 ],
 "description": "Vectors, matrices and the core ideas of linear algebra.",
 "tags": [
  "linear algebra",
  "matrices",
  "vectors"
 ],
 "frontLabel": "Term",
 "backLabel": "Meaning",
 "cards": [
  [
   "Scalar",
   "A single number, as opposed to a vector"
  ],
  [
   "Vector",
   "An ordered list of numbers; has magnitude and direction"
  ],
  [
   "Magnitude (norm) of a vector",
   "Its length, √(x² + y² + ...)"
  ],
  [
   "Unit vector",
   "A vector of length 1"
  ],
  [
   "Zero vector",
   "The vector whose entries are all 0"
  ],
  [
   "Vector addition",
   "Add matching entries"
  ],
  [
   "Scalar multiplication",
   "Multiply every entry of a vector or matrix by the same number"
  ],
  [
   "Dot product",
   "Sum of products of matching entries; a · b = |a||b| cos θ"
  ],
  [
   "Perpendicular vectors",
   "Their dot product is 0 (orthogonal)"
  ],
  [
   "Cross product",
   "A vector perpendicular to two 3D vectors; length |a||b| sin θ"
  ],
  [
   "Linear combination",
   "A sum of vectors each multiplied by a scalar"
  ],
  [
   "Span",
   "The set of all linear combinations of some vectors"
  ],
  [
   "Linear independence",
   "No vector in the set is a linear combination of the others"
  ],
  [
   "Linear dependence",
   "At least one vector can be built from the others"
  ],
  [
   "Basis",
   "A linearly independent set of vectors that spans a space"
  ],
  [
   "Dimension",
   "The number of vectors in a basis"
  ],
  [
   "Standard basis of R²",
   "(1, 0) and (0, 1)"
  ],
  [
   "Vector space",
   "A set where vectors can be added and scaled by numbers following set rules"
  ],
  [
   "Subspace",
   "A vector space inside a larger one"
  ],
  [
   "Matrix",
   "A rectangular array of numbers in rows and columns"
  ],
  [
   "Order (size) of a matrix",
   "Rows × columns, such as 2 × 3"
  ],
  [
   "Square matrix",
   "A matrix with the same number of rows and columns"
  ],
  [
   "Identity matrix",
   "A square matrix with 1s on the main diagonal and 0s elsewhere; AI = A"
  ],
  [
   "Zero matrix",
   "A matrix with all entries 0"
  ],
  [
   "Diagonal matrix",
   "A square matrix with zeros off the main diagonal"
  ],
  [
   "Symmetric matrix",
   "A matrix equal to its transpose"
  ],
  [
   "Transpose",
   "Swap rows and columns; written Aᵀ"
  ],
  [
   "Matrix addition",
   "Add matching entries; matrices need the same size"
  ],
  [
   "Matrix multiplication rule",
   "An m × n matrix times an n × p matrix gives an m × p matrix"
  ],
  [
   "Is matrix multiplication commutative?",
   "No. AB is usually not equal to BA"
  ],
  [
   "Inverse matrix",
   "A⁻¹ such that A A⁻¹ = I"
  ],
  [
   "Invertible (nonsingular) matrix",
   "A square matrix with an inverse; its determinant is not 0"
  ],
  [
   "Singular matrix",
   "A matrix with no inverse; determinant 0"
  ],
  [
   "Determinant of a 2×2 matrix [a b; c d]",
   "ad − bc"
  ],
  [
   "Determinant meaning",
   "Scale factor of area or volume under the matrix's transformation"
  ],
  [
   "Trace of a matrix",
   "The sum of the diagonal entries"
  ],
  [
   "Rank",
   "The number of linearly independent rows (or columns)"
  ],
  [
   "Null space",
   "All vectors x with Ax = 0"
  ],
  [
   "Column space",
   "The span of the columns of a matrix"
  ],
  [
   "Row space",
   "The span of the rows of a matrix"
  ],
  [
   "Linear transformation",
   "A function T where T(u + v) = T(u) + T(v) and T(cu) = cT(u)"
  ],
  [
   "System of linear equations as a matrix",
   "Ax = b"
  ],
  [
   "Augmented matrix",
   "The coefficient matrix with the constants column attached"
  ],
  [
   "Gaussian elimination",
   "Using row operations to turn a matrix into row-echelon form"
  ],
  [
   "Row operations",
   "Swap rows, multiply a row by a nonzero number, add a multiple of one row to another"
  ],
  [
   "Row-echelon form",
   "A staircase form with leading entries moving to the right"
  ],
  [
   "Reduced row-echelon form",
   "Row-echelon form with leading 1s and zeros above and below them"
  ],
  [
   "Pivot",
   "A leading nonzero entry in a row used in elimination"
  ],
  [
   "Eigenvalue",
   "A number λ such that Av = λv for some nonzero vector v"
  ],
  [
   "Eigenvector",
   "A nonzero vector that only gets scaled by a matrix"
  ],
  [
   "Characteristic polynomial",
   "det(A − λI); its roots are the eigenvalues"
  ],
  [
   "Orthogonal matrix",
   "A square matrix whose columns are perpendicular unit vectors; Qᵀ = Q⁻¹"
  ],
  [
   "Projection",
   "The shadow of one vector onto another"
  ],
  [
   "Rotation matrix in 2D",
   "[cos θ, −sin θ; sin θ, cos θ]"
  ],
  [
   "Cramer's rule",
   "Solve a system using ratios of determinants"
  ],
  [
   "Orthonormal basis",
   "A basis of perpendicular unit vectors"
  ],
  [
   "Gram-Schmidt process",
   "A method to turn a basis into an orthonormal one"
  ],
  [
   "Diagonalization",
   "Writing A = PDP⁻¹ with D diagonal"
  ],
  [
   "Singular value decomposition (SVD)",
   "Factoring a matrix into rotations and stretches: A = UΣVᵀ"
  ]
 ]
}