MATH1048 Linear Algebra I Lecture Notes Study Guide

Definition of Complex Numbers: A complex number is defined as $z = a + bi$ where $a, b ext{ are real numbers, } i ext{ is the imaginary unit with } i^2 = -1.$
Operations with Complex Numbers:
- Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$.
- Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$.
- Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.
Graphical Representation: Complex numbers can be represented as points $(a, b)$ in the Cartesian plane, with the x-axis as the real part and the y-axis as the imaginary part.
Modulus: The modulus of a complex number $z = a + bi$ is defined as $|z| = \sqrt{a^2 + b^2}$.
Argument: The argument $ ext{arg}(z)$ is the angle $ heta$ in polar coordinates from the positive x-axis, defined up to $2 ext{π}$.

Definition of Real n-space: Denoted as $R^n$, this is the set of all n-tuples where each element is a real number.
Scalar Product: Defined as $u ullet v = u1v1 + u2v2 + … + unvn$, yielding a real number.
Norm of a Vector: The norm is $||v|| = \sqrt{v1^2 + v2^2 + … + v_n^2}$.
Projections and Cauchy-Schwarz Inequality: Relates projections of vectors and establishes that the absolute value of the scalar product does not exceed the product of their magnitudes.
Various equations of lines and planes in $R^3$: Both parametric and Cartesian forms discussed, with illustrations.

Definition of Matrix: An $m \times n$ matrix is an array of real numbers.
Operations with Matrices: Multiplication by a scalar, addition, subtraction, and matrix multiplication are defined and explored in depth.
Transpose and Inverse of a Matrix: Definitions and properties—emphasizing criteria for invertibility based on determinants.
- For a matrix to be invertible, $det(A) \neq 0$.
- The inverse is $A^{-1} = (1/det(A)) ext{adj}(A)$.
Rank of a Matrix: The rank is defined as the number of leading 1's in the row echelon form of the matrix.

Definition of a Linear System: A collection of linear equations in matrix form $A x = b$.
Row Operations: Augmented matrices can be manipulated using elementary row operations.
Gaussian Elimination: Method for solving systems of linear equations by transforming the matrix to reduced row echelon form (RREF).
Homogeneous Systems: Characterization of consistent and inconsistent systems based on their row echelon forms.
Rank and Solutions: The relationship between rank, number of variables, and the existence of solutions is established.

Axiomatic Definition: Formal definition of determinants through linearity and other properties.
Determinant Properties: Including how the determinant behaves under elementary row operations and for specific matrix forms (upper triangular).
Cofactor Expansion: Method for calculating the determinant using minors and cofactors.

Definition of Linear Transformations: Maps that preserve addition and scalar multiplication.
Matrix Representation of Linear Transformations: Establishing the correspondence between linear transformations and matrices.
Composition and Inverse of Linear Transformations: Rules and properties surrounding compositional transformations and finding inverses.

Subspace Definition and Examples: Exploration of subspaces, null space definitions, and conditions that establish a subspace.
Linear Combinations and Span: Analysis of spanning sets and their relationship to subspaces.
Linear Independence and Dependence: Conditions that determine when a set of vectors is considered independent.

Definition of Eigenvalues and Eigenvectors: Understanding their role within matrix formulations and problems.
Characteristic Polynomial: Method to find eigenvalues via determinants of $A - \lambda I$.
Finding Eigenvectors for Given Eigenvalues: Implements procedures to calculate eigenvectors after determining the eigenvalues.

Definitions of Orthogonal and Orthonormal Sets: The characteristics that define orthogonality in a vector space.
Gram-Schmidt Process: An algorithm for converting a set of linearly independent vectors into an orthonormal set.
Quasi-Quadratic Forms and Conic Sections: Informs how quadratic forms correspond to geometric representations in $R^n$.

Various examples illustrating concepts such as the effect of transformations, bases in subspaces, and determinant calculation through cofactor expansion.

Students should master the definitions and theorem applications, especially in the context of solving linear systems and eigenvalue problems, as these are foundational to the understanding of higher mathematics, particularly in Linear Algebra II and applications in science and engineering.