Advanced Algebra

 

1. Introduction to Advanced Algebra

  • Definition: Advanced Algebra builds on the foundational concepts of Elementary Algebra, focusing on more complex equations, functions, and systems. It is essential for understanding higher-level mathematics like calculus, linear algebra, and abstract algebra.

2. Polynomial Functions

  • Polynomials: Expressions involving sums of powers of variables with coefficients.
    • Standard Form: anxn+an1xn1++a1x+a0a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0where an0
    • Degree of a Polynomial: The highest power of the variable (e.g., 3x4+2x3+x+53x^4 + 2x^3 + x + 5 has a degree of 4).
  • Operations:
    • Addition/Subtraction: Combine like terms.
    • Multiplication: Distribute each term.
    • Division: Long division or synthetic division.
  • Factoring: Breaking down polynomials into products of lower-degree polynomials.
    • Methods: Common factor, grouping, difference of squares, sum/difference of cubes, and quadratic trinomials.

3. Rational Functions

  • Definition: A ratio of two polynomials, f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)} where P(x)P(x) and Q(x)Q(x) are polynomials.
  • Domain: The set of all real numbers except where the denominator Q(x)Q(x) is zero.
  • Asymptotes:
    • Vertical Asymptote: Occurs where the denominator is zero.
    • Horizontal Asymptote: Determined by the degrees of the polynomials in the numerator and denominator.
    • Oblique Asymptote: Occurs when the degree of the numerator is exactly one more than the degree of the denominator.

4. Complex Numbers

  • Definition: Numbers in the form a+ib, where ii is the imaginary unit (i2=1i^2 = -1).
  • Operations:
    • Addition/Subtraction: Combine real and imaginary parts separately.
    • Multiplication: Use the distributive property and i2=1i^2 = -1
    • Division: Multiply the numerator and denominator by the conjugate of the denominator.
  • Polar Form: Representing complex numbers as r(cosθ+isinθ)r(\cos \theta + i \sin \theta) or reiθr e^{i\theta}, where r=a2+b2r = \sqrt{a^2 + b^2} and θ=tan1(ba)
  • De Moivre’s Theorem: [r(cosθ+isinθ)]n=rn(cosnθ+isinnθ)[r(\cos \theta + i \sin \theta)]^n = r^n (\cos n\theta + i \sin n\theta), useful for finding powers and roots of complex numbers.

5. Exponential and Logarithmic Functions

  • Exponential Functions: Functions of the form f(x)=abxf(x) = a \cdot b^x, where b>0b > 0 and b1b \neq 1.
    • Properties:
      • bm×bn=bm+nb^m \times b^n = b^{m+n}
      • bmbn=bmn\frac{b^m}{b^n} = b^{m-n}
      • (bm)n=bmn
  • Logarithmic Functions: The inverse of exponential functions, y=logb(x) means by=xb^y = x
    • Properties:
      • logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)
      • logb(xy)=logb(x)logb(y)
      • logb(xy)=ylogb(x)\log_b(x^y) = y \log_b(x)
  • Natural Logarithm: ln(x)\ln(x) is the logarithm to the base ee (Euler’s number, approximately 2.718).
  • Change of Base Formula: logb(x)=logc(x)logc(b)

6. Systems of Equations and Inequalities

  • Linear Systems:
    • Solving Methods: Substitution, elimination, matrix methods (e.g., Gaussian elimination), and graphical methods.
    • Number of Solutions: Consistent and independent (one solution), consistent and dependent (infinitely many solutions), or inconsistent (no solution).
  • Nonlinear Systems:
    • Include equations like circles, parabolas, and ellipses.
    • Solving Methods: Substitution, elimination, and graphical methods.
  • Systems of Inequalities:
    • Graphical Solutions: Graph each inequality and find the intersection (solution region).
    • Linear Programming: Optimizing a linear objective function subject to a system of linear inequalities.

7. Matrices and Determinants

  • Matrix Operations:
    • Addition/Subtraction: Add or subtract corresponding elements.
    • Multiplication: The dot product of rows and columns.
    • Determinants: A scalar value that can be computed from the elements of a square matrix, useful for solving systems of linear equations.
      • Properties: If the determinant is zero, the matrix is singular and has no inverse.
  • The inverse of a Matrix: A1           is the matrix that satisfies AA1=IA \cdot A^{-1} = I, where II is the identity matrix.
  • Cramer’s Rule: A method for solving linear systems using determinants.

8. Sequences and Series

  • Sequences: An ordered list of numbers defined by a rule.
    • Arithmetic Sequence: The difference between consecutive terms is constant. an=a1+(n1)d
    • Geometric Sequence: The ratio between consecutive terms is constant. an=a1rn1a_n = a_1 \cdot r^{n-1}
  • Series: The sum of the terms of a sequence.
    • Arithmetic Series: Sum of an arithmetic sequence, Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n).
    • Geometric Series: Sum of a geometric sequence, Sn=a11rn1rS_n = a_1 \cdot \frac{1-r^n}{1-r} for r1.
  • Infinite Series: Sum of an infinite sequence. Converges if the sum approaches a finite limit.
    • Convergence Tests: Tests like the ratio and root tests determine whether an infinite series converges or diverges.

9. Binomial Theorem

  • Statement: Describes the algebraic expansion of powers of a binomial. (x+y)n=k=0n(nk)xnkyk(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k, where (nk)\binom{n}{k}is a binomial coefficient.
  • Application: Useful in expanding polynomials and in probability theory.

10. Conic Sections

  • Definition: Curves obtained by intersecting a cone with a plane.
  • Types:
    • Circle: x2+y2=r2x^2 + y^2 = r^2
    • Ellipse: x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
    • Parabola: y=ax2+bx+c
    • Hyperbola: x2a2y2b2=1
  • Focus-Directrix Property: Each conic section has a focus and a directrix, and the distance from any point on the curve to the focus is proportional to its distance to the directrix.

11. Vectors and Vector Spaces

  • Vectors: Quantities with both magnitude and direction. Represented as v=v1,v2,,vn\mathbf{v} = \langle v_1, v_2, \dots, v_n \rangle
  • Operations:
    • Addition/Subtraction: Component-wise addition or subtraction.
    • Scalar Multiplication: Multiplying each component by a scalar.
    • Dot Product: uv=u1v1+u2v2++unvn \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \dots + u_nv_n   gives a scalar.
    • Cross Product: Only defined in three dimensions, gives a vector perpendicular to the plane containing the original vectors.
  • Vector Spaces: A set of vectors closed under addition and scalar multiplication, with a defined zero vector.

12. Linear Transformations

  • Definition: Functions that map vectors to vectors, preserving vector addition and scalar multiplication.
  • Matrix Representation: Any linear transformation can be represented by a matrix AA, where T(v)=AvT(\mathbf{v}) = A\mathbf{v}
  • Eigenvalues and Eigenvectors: For a matrix AA, if Av=λvA\mathbf{v} = \lambda\mathbf{v}, then λ\lambda is an eigenvalue and v\mathbf{v} is an eigenvector.
  • Diagonalization: The process of finding a diagonal matrix DD similar to a given matrix AA, such that A=PDP1A = PDP^{-1}



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