Math 54 (UC Berkeley) Flashcards

Study Notes

Key Concepts in Linear Algebra

Orthogonality: Defined by the equation ‖u+v‖² = ‖u‖² + ‖v‖², indicating that two vectors u and v are orthogonal if their dot product u₀v = 0.
Orthogonal Complement: A vector x belongs to the orthogonal complement of a subspace W if it is orthogonal to every vector in the set that spans W, making it also a subspace of Rⁿ.
Orthogonal Complements of Row and Column Spaces:
- OC of Row A is represented by Nul A.
- OC of Col A is represented by the transpose of Nul A.
Inner Product: Given by the formula u₀v = ‖u‖‖v‖cosθ, representing the cosine of the angle between two vectors multiplied by their magnitudes.
Orthogonal Basis and Linear Combinations: Any vector y can be expressed as a linear combination of an orthogonal basis {u₁, u₂,...} where c₁ = (y⊗u₁)/(u₁⊗u₁).

Orthogonal Decomposition and Projections

Orthogonal Decomposition Theorem: States that any vector y in Rⁿ can be decomposed into components within a subspace W and orthogonal to it, represented as y = ŷ + z.
Orthogonal Projection: The projection of a vector y onto a line L defined by a vector u is given as ŷ = (y₀u)/(u₀u)u.

Matrices and Their Properties

Orthonormal Matrices: Matrices that satisfy UTU = I and maintain norms, preserving the dot product, where UT is the transpose of U and U∧-1 is its inverse.
QR Factorization: Any matrix A can be factored as A = QR, where Q is an orthonormal matrix and R = QTA.

Least Squares and Eigenvalues

Least Squares Methods: The solution to the least squares problem is given by the equation Rx = QT b, where A is involved in projecting onto the column space.
Cauchy-Schwarz Inequality: States that for any vectors u and v, the absolute value of their inner product is less than or equal to the product of their magnitudes, ||u₀v|| ≤ ‖u‖‖v‖.
Eigenvalues and Symmetric Matrices: Symmetric matrices (A = AT) have real eigenvalues, and eigenvectors corresponding to different eigenvalues are orthogonal.

Properties and Theorems

Invertible Matrix Theorem: Outlines 22 equivalent conditions, such as having n pivot positions, being row equivalent to an nxn matrix, and that Ax = 0 has only the trivial solution.
Rank: Defined as the number of non-zero rows after a matrix is put into row-echelon form, with the rank theorem stating that rank + nullity equals the number of columns of matrix A.

Fourier Series Representation

Fourier Series: The expansion of a function f(x) can be represented as a₀/2 + Σ a₁cos(nπx/L) + b₁sin(nπx/L), where coefficients are computed from integrals over defined bounds.
Fourier Coefficients:
- a₀ is computed as (1/L)∫f(x)dx over bounds (-L, L).
- a₁ is defined as (2/L)∫f(x)cos(nπx/L)dx over bounds (0, L).
- b₁ is defined as (2/L)∫f(x)sin(nπx/L)dx over bounds (0, L).

Solutions to Differential Equations

Solving x' = Ax with Eigenvalues: The general solution can be expressed as x(t) = c₁e^(λt)u + c₂e^(λt)v, where u and v are the eigenvectors corresponding to eigenvalues λ.
Fundamental Solution Set: The set {e^(λ₁t)u₁, e^(λ₂t)u₂, ...} forms the basis for solutions to the equation x' = Ax.

Complex Eigenvalues

Complex Eigenvalues: For eigenvalues given by α + βi, the corresponding eigenvectors are a + bi, leading to a solution that combines exponential and trigonometric functions involving cosines and sines.

Description

Explore essential concepts in linear algebra with these flashcards from Math 54 at UC Berkeley. This quiz covers topics such as orthogonality, orthogonal complements, and linear combinations, helping students grasp crucial definitions and properties. Perfect for students preparing for exams or wishing to reinforce their understanding of vector spaces.