➗Linear Algebra and Differential Equations Unit 1 – Linear Systems and Matrices

Key Concepts and Definitions

• Linear systems represent a set of linear equations with multiple variables
• Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns
• A matrix element $a_{ij}$ is the entry in the $i$-th row and $j$-th column of matrix $A$
• Matrix addition and subtraction require matrices to have the same dimensions and involve element-wise operations
• Matrix multiplication is a binary operation that produces a matrix from two matrices, following specific rules
  • The number of columns in the first matrix must equal the number of rows in the second matrix
  • The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix
• Scalar multiplication involves multiplying each element of a matrix by a scalar value
• The identity matrix, denoted as $I_n$, is a square matrix with ones on the main diagonal and zeros elsewhere
• The inverse of a square matrix $A$, denoted as $A^{-1}$, is a matrix such that $AA^{-1} = A^{-1}A = I$

Linear Systems and Their Properties

• A linear system is a collection of linear equations involving the same set of variables
• The solution to a linear system is an assignment of values to the variables that satisfies all the equations simultaneously
• A linear system can have a unique solution, infinitely many solutions, or no solution
• The number of equations and the number of variables in a linear system determine its properties
  • If the number of equations is less than the number of variables, the system is underdetermined and has infinitely many solutions or no solution
  • If the number of equations is equal to the number of variables, the system can have a unique solution or no solution
  • If the number of equations is greater than the number of variables, the system is overdetermined and has no solution or a unique solution (if the equations are consistent)
• Gaussian elimination is a method for solving linear systems by transforming the augmented matrix into row echelon form
• Back-substitution is used to find the values of variables in a linear system once it is in row echelon form
• Consistency of a linear system refers to the existence of a solution
  • A consistent system has at least one solution (unique or infinitely many)
  • An inconsistent system has no solution

Matrix Operations and Algebra

• Matrix addition is commutative: $A + B = B + A$
• Matrix addition is associative: $(A + B) + C = A + (B + C)$
• The zero matrix, denoted as $0$, is a matrix with all elements equal to zero and serves as the additive identity: $A + 0 = A$
• Matrix subtraction is defined as the addition of a matrix and the negative of another matrix: $A - B = A + (-B)$
• Matrix multiplication is associative: $(AB)C = A(BC)$
• Matrix multiplication is distributive over matrix addition: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$
• The identity matrix serves as the multiplicative identity: $AI_n = I_nA = A$
• Matrix multiplication is not commutative in general: $AB \neq BA$
• The transpose of a matrix $A$, denoted as $A^T$, is obtained by interchanging its rows and columns
  • $(A^T)^T = A$
  • $(A + B)^T = A^T + B^T$
  • $(AB)^T = B^TA^T$

Solving Linear Systems with Matrices

• A linear system can be represented using an augmented matrix, which combines the coefficient matrix and the constant terms
• Elementary row operations can be applied to the augmented matrix to solve the linear system
  • Swap the positions of two rows
  • Multiply a row by a non-zero scalar
  • Add a multiple of one row to another row
• Gaussian elimination involves applying elementary row operations to transform the augmented matrix into row echelon form
  • In row echelon form, all leading coefficients (i.e., the leftmost non-zero entry in each row) are equal to 1, and the column containing the leading coefficient of a row has zeros in all other entries
• Reduced row echelon form is a unique matrix form obtained by further applying Gaussian elimination to the row echelon form
  • In reduced row echelon form, the leading coefficient in each row is 1, and the column containing the leading 1 has zeros in all other entries
• The rank of a matrix is the number of non-zero rows in its reduced row echelon form
  • A linear system has a unique solution if and only if the rank of the augmented matrix is equal to the rank of the coefficient matrix and the number of variables
• Cramer's rule is a formula for solving linear systems using determinants, applicable when the system has a unique solution

Determinants and Their Applications

• The determinant is a scalar value associated with a square matrix, denoted as $det(A)$ or $|A|$
• The determinant of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is calculated as $det(A) = ad - bc$
• The determinant of a 3x3 matrix can be calculated using the Laplace expansion or Sarrus' rule
• Properties of determinants:
  • The determinant of the identity matrix is 1: $det(I_n) = 1$
  • The determinant of a matrix is equal to the determinant of its transpose: $det(A) = det(A^T)$
  • If a matrix has a row or column of zeros, its determinant is zero
  • Interchanging two rows or columns of a matrix changes the sign of its determinant
  • Multiplying a row or column of a matrix by a scalar $k$ multiplies the determinant by $k$
• The determinant can be used to check if a matrix is invertible
  • A square matrix $A$ is invertible if and only if $det(A) \neq 0$
• Cramer's rule uses determinants to solve linear systems with unique solutions
• The determinant can be used to calculate the area of a parallelogram or the volume of a parallelepiped in higher dimensions

Vector Spaces and Subspaces

• A vector space is a set $V$ of elements called vectors, along with two operations (addition and scalar multiplication) that satisfy certain axioms
  • Closure under addition and scalar multiplication
  • Associativity of addition and scalar multiplication
  • Commutativity of addition
  • Existence of the zero vector and additive inverses
  • Existence of the scalar multiplicative identity
  • Distributivity of scalar multiplication over vector addition and field addition
• Examples of vector spaces include $\mathbb{R}^n$, the set of all $n$-tuples of real numbers, and the set of all $m \times n$ matrices with real entries
• A subspace is a subset of a vector space that is itself a vector space under the same operations
• To verify if a subset is a subspace, check if it is closed under addition and scalar multiplication and contains the zero vector
• The intersection of two subspaces is always a subspace
• The union of two subspaces is a subspace if and only if one subspace is contained within the other
• The span of a set of vectors is the smallest subspace containing all linear combinations of those vectors
• A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others
• A basis is a linearly independent set of vectors that spans the entire vector space
• The dimension of a vector space is the number of vectors in its basis

Linear Transformations

• A linear transformation (or linear map) is a function $T: V \rightarrow W$ between two vector spaces $V$ and $W$ that satisfies the following properties:
  • Additivity: $T(u + v) = T(u) + T(v)$ for all $u, v \in V$
  • Homogeneity: $T(cu) = cT(u)$ for all $u \in V$ and scalar $c$
• The kernel (or null space) of a linear transformation $T$ is the set of all vectors $v \in V$ such that $T(v) = 0$
  • The kernel is always a subspace of the domain $V$
• The range (or image) of a linear transformation $T$ is the set of all vectors $T(v)$ for $v \in V$
  • The range is always a subspace of the codomain $W$
• A linear transformation can be represented by a matrix $A$ such that $T(x) = Ax$ for all $x \in V$
• The matrix representation of a linear transformation depends on the chosen bases for the domain and codomain
• Composition of linear transformations corresponds to matrix multiplication of their representative matrices
• An isomorphism is a bijective linear transformation between two vector spaces
  • Two vector spaces are isomorphic if there exists an isomorphism between them
  • Isomorphic vector spaces have the same dimension

Real-World Applications and Examples

• Linear systems can model various real-world problems, such as:
  • Balancing chemical equations in chemistry
  • Analyzing electrical circuits using Kirchhoff's laws
  • Solving network flow problems in operations research
• Matrices have numerous applications, including:
  • Representing and manipulating images in computer graphics
  • Analyzing social networks and web page rankings (e.g., Google's PageRank algorithm)
  • Modeling population dynamics and ecological systems using Leslie matrices
• Markov chains, which use stochastic matrices to model systems that transition between states, have applications in:
  • Natural language processing and speech recognition
  • Financial modeling and market analysis
  • Biology and genetics (e.g., DNA sequence analysis)
• Linear transformations are used in:
  • Computer graphics and geometric modeling (e.g., rotations, reflections, and scaling)
  • Quantum mechanics to represent physical observables and states
  • Machine learning and data analysis (e.g., principal component analysis and dimensionality reduction)
• Eigenvalues and eigenvectors, which are closely related to linear transformations, have applications in:
  • Vibration analysis and structural engineering
  • Image compression and facial recognition
  • Stability analysis of dynamical systems and differential equations