📡Bioengineering Signals and Systems Unit 2 – Linear Algebra & Complex Numbers Foundations

Key Concepts

• Linear algebra studies vector spaces and linear mappings between them
• Vectors represent quantities with both magnitude and direction (force, velocity)
• Matrices are rectangular arrays of numbers used to represent linear transformations
• Complex numbers extend the real number system by introducing the imaginary unit $i$ where $i^2 = -1$
• Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ connects complex numbers with trigonometry
• Linear transformations map vectors from one vector space to another while preserving linearity
• Eigenvalues and eigenvectors characterize the behavior of linear transformations
• Signal processing involves analyzing, modifying, and synthesizing signals using linear algebra techniques

Vector and Matrix Operations

• Vector addition and subtraction are performed element-wise (component-wise)
• Scalar multiplication of a vector scales each component by the scalar
• Dot product (inner product) of two vectors $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n$
  • Measures the similarity or projection of one vector onto another
• Cross product of two 3D vectors $\mathbf{a} \times \mathbf{b}$ results in a vector perpendicular to both
  • Magnitude equals the area of the parallelogram formed by the vectors
• Matrix multiplication is performed by multiplying rows of the first matrix with columns of the second
  • Resulting matrix has dimensions (rows of first) $\times$ (columns of second)
• Matrix transpose $A^T$ interchanges the rows and columns of matrix $A$
• Identity matrix $I$ has ones on the main diagonal and zeros elsewhere, $AI = IA = A$

Complex Numbers and Euler's Formula

• Complex numbers have the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part
• Complex conjugate of $a + bi$ is $a - bi$, product of a complex number and its conjugate is real
• Complex numbers can be represented as points on the complex plane with real and imaginary axes
• Polar form of a complex number is $r(\cos\theta + i\sin\theta)$, where $r$ is magnitude and $\theta$ is angle
• Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ relates exponential function to trigonometric functions
  • Allows expressing complex numbers as $re^{i\theta}$ in exponential form
• De Moivre's formula $(r(\cos\theta + i\sin\theta))^n = r^n(\cos n\theta + i\sin n\theta)$ simplifies complex exponentiation
• Complex numbers are used in signal processing to represent sinusoidal signals and frequency components

Linear Transformations

• Linear transformations $T$ satisfy $T(a\mathbf{u} + b\mathbf{v}) = aT(\mathbf{u}) + bT(\mathbf{v})$ for scalars $a, b$ and vectors $\mathbf{u}, \mathbf{v}$
• Can be represented by matrices, transforming a vector is equivalent to matrix-vector multiplication
• Composition of linear transformations corresponds to matrix multiplication
• Kernel (null space) of a linear transformation is the set of vectors mapped to the zero vector
• Range (image) of a linear transformation is the set of all possible output vectors
• Rank of a matrix is the dimension of its range, nullity is the dimension of its kernel
  • Rank-nullity theorem states that rank + nullity = number of columns

Eigenvalues and Eigenvectors

• Eigenvector $\mathbf{v}$ of a matrix $A$ satisfies $A\mathbf{v} = \lambda\mathbf{v}$ for some scalar $\lambda$
  • $\lambda$ is the corresponding eigenvalue, measures the scaling factor of the eigenvector
• Eigenvalues are roots of the characteristic polynomial $\det(A - \lambda I) = 0$
• Eigenvectors corresponding to distinct eigenvalues are linearly independent
• Diagonalizable matrices have a full set of linearly independent eigenvectors
  • Can be factored as $A = PDP^{-1}$, where $D$ is diagonal with eigenvalues, $P$ has eigenvectors as columns
• Eigendecomposition allows efficient computation of matrix powers $A^n = PD^nP^{-1}$
• Symmetric matrices have real eigenvalues and orthogonal eigenvectors

Applications in Signal Processing

• Signals can be represented as vectors, with each component corresponding to a time point or frequency
• Linear time-invariant (LTI) systems are described by linear transformations
  • Impulse response fully characterizes the system's behavior
• Convolution of input signal with impulse response computes the output signal
  • Equivalent to matrix-vector multiplication in discrete-time
• Fourier transform decomposes a signal into its frequency components using complex exponentials
  • Represents the signal in the frequency domain
• Eigenanalysis of covariance matrices is used in principal component analysis (PCA) for dimensionality reduction
  • Eigenvectors capture the directions of maximum variance in the data
• Singular value decomposition (SVD) factorizes a matrix into orthogonal matrices and a diagonal matrix
  • Used for noise reduction, data compression, and feature extraction

Problem-Solving Techniques

• Break down complex problems into smaller, more manageable subproblems
• Identify the key concepts and techniques relevant to the problem
• Visualize the problem geometrically, using vector spaces and transformations
• Exploit the properties of matrices and vectors to simplify computations
  • Utilize matrix decompositions (eigendecomposition, SVD) when appropriate
• Apply linear algebra theorems and identities to derive solutions
  • Rank-nullity theorem, Cayley-Hamilton theorem, properties of eigenvalues and eigenvectors
• Verify the correctness of the solution by checking against known properties or special cases
• Interpret the results in the context of the original problem and its physical significance

Connections to Bioengineering

• Biomedical signals (ECG, EEG, EMG) are analyzed using linear algebra techniques
  • Filtering, feature extraction, and pattern recognition
• Biomechanical systems are modeled using vectors and matrices
  • Forces, velocities, and accelerations of body segments
• Medical imaging (CT, MRI) relies on linear transformations and matrix operations
  • Image reconstruction, registration, and segmentation
• Pharmacokinetic models use linear differential equations to describe drug absorption and elimination
  • Eigenvalues determine the stability and time constants of the system
• Genetic regulatory networks are represented by matrices capturing gene interactions
  • Eigenvectors correspond to stable gene expression patterns
• Principal component analysis (PCA) is used to identify patterns in gene expression data
  • Reduces the dimensionality of high-throughput genomic datasets
• Singular value decomposition (SVD) is applied in protein structure analysis and drug discovery
  • Identifies key structural features and potential drug targets