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Precalculus - Vectors, Complex Numbers, Linear Systems, Matrices, and Determinants

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Termini in questo insieme (25)

  • Definition of a vector in the plane
    A vector in the plane is a directed line segment with magnitude and direction, represented algebraically as an ordered pair (x, y).
  • Vector addition in R2
    Sum of vectors (a, b) + (c, d) = (a + c, b + d), commutative and associative.
  • Scalar multiplication of vectors
    For vector (a, b) and scalar α, α(a, b) = (αa, αb).
  • Dot product of vectors in R2
    For u = (u1, u2) and v = (v1, v2), u · v = u1v1 + u2v2, a scalar.
  • Angle between two vectors
    cos θ = (u · v) / (∥u∥ ∥v∥), where θ is the angle between nonzero vectors u and v.
  • Complex number standard form
    z = a + bi, where a, b ∈ R and i² = -1.
  • Complex conjugate and modulus
    Conjugate of z = a + bi is z̄ = a - bi; modulus |z| = √(a² + b²).
  • Multiplication of complex numbers
    (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
  • Polar (modulus-argument) form of complex numbers
    z = r(cos θ + i sin θ) = r e^{iθ}, where r = |z| and θ = arg z.
  • De Moivre's Theorem
    (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) for integer n.
  • nth roots of a complex number
    The n distinct nth roots of z = r e^{iθ} are z_k = r^{1/n} e^{i(θ + 2πk)/n}, k = 0,...,n-1.
  • System of linear equations matrix form
    AX = B, where A is coefficient matrix, X vector of unknowns, B vector of constants.
  • Elementary row operations
    Operations: multiply row by nonzero scalar, interchange rows, add multiple of one row to another.
  • Reduced row-echelon form
    Matrix form with leading 1s, zeros below and above leading 1s, zero rows at bottom.
  • Matrix addition and scalar multiplication
    Add matrices element-wise; multiply each element by scalar.
  • Matrix multiplication condition
    Product AB defined if columns of A = rows of B; result is matrix with rows of A and columns of B.
  • Transpose of a matrix
    Transpose AT of matrix A is obtained by swapping rows and columns.
  • Invertible matrix and inverse
    Square matrix A is invertible if ∃ B such that AB = BA = I; B = A^{-1}.
  • Determinant of 2x2 matrix
    det(A) = ad - bc for A = [[a, b], [c, d]].
  • Cofactor expansion for determinant
    det(A) = sum of elements of any row or column times their cofactors.
  • Cramer's Rule
    Solution x_j = det(A_j) / det(A), where A_j replaces j-th column of A with constants vector.
  • Vector equation of a line in R3
    r = a + λd, where a is a point on the line and d is a direction vector.
  • Dot product and angle between vectors
    a · b = ∥a∥∥b∥ cos θ, where θ is the angle between vectors a and b.
  • Cross product of vectors in R3
    a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1), orthogonal to both a and b.
  • Equation of a plane in R3
    n · (r - a) = 0 or n1x + n2y + n3z = d, where n is normal vector and a is a point on the plane.