Vectors and Matrices — Study Notes and Flashcards Flashcards

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Row vector

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A 1×n array with elements arranged horizontally. It contrasts with a column vector and is useful for representing a single observation or a transposed column.

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Column vector

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An n×1 array with elements arranged vertically. Column vectors are commonly used to represent variables or coefficients in linear systems.

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Indexing

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Accessing individual elements of a vector or matrix using integer positions. Indexing can be 0-based or 1-based depending on the language and is the basis for element retrieval and assignment.

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Slicing

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Selecting ranges or subblocks of elements from vectors or matrices using range notation. Slicing produces subvectors or submatrices for further operations or analysis.

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linspace

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Creates a vector of linearly spaced values between two endpoints. Common signature is linspace(a,b,n) which returns n evenly spaced points from a to b.

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logspace

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Generates logarithmically spaced values between powers of ten. logspace(a,b,n) produces n points between $10^a$ and $10^b$, useful for log-scale sampling.

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Matrix addition

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Elementwise addition of two matrices of the same shape. Each resulting element equals the sum of corresponding elements from the addends.

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Matrix multiplication

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Combines rows of the first matrix with columns of the second via dot products; defined when inner dimensions match. Matrix multiplication is not commutative in general and yields linear combinations of columns or rows.

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Transpose

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Operation that flips a matrix across its diagonal, swapping rows and columns. Denoted $A^T$, it converts row vectors to column vectors and vice versa.

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Inverse

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For a square matrix $A$, the inverse $A^{-1}$ satisfies $AA^{-1}=I$ when $A$ is nonsingular. Computing the inverse directly is often less stable than solving systems via decompositions.

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Determinant

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A scalar value $\det(A)$ that indicates volume scaling and singularity; if zero, the matrix is singular. Determinants are used to test invertibility but are sensitive to numerical errors for large matrices.

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Rank

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The number of linearly independent rows or columns of a matrix, denoted $\operatorname{rank}(A)$. Rank determines solvability and uniqueness of linear systems.

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Trace

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The sum of a square matrix's diagonal entries, $\operatorname{trace}(A)=\sum_i A_{ii}$. Trace is invariant under similarity transformations and relates to eigenvalues (sum of eigenvalues).

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Eigenvalue

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A scalar $\lambda$ for which there exists a nonzero vector $v$ with $Av=\lambda v$. Eigenvalues characterize intrinsic modes and scaling factors of a linear transform.

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Eigenvector

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A nonzero vector $v$ that satisfies $Av=\lambda v$ for some eigenvalue $\lambda$. Eigenvectors point in directions that are invariant under the linear map up to scaling.

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eig

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A function or routine that computes eigenvalues and eigenvectors of a matrix. eig typically returns a vector of eigenvalues and a matrix whose columns are eigenvectors.

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zeros

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Constructor that creates a matrix filled with zeros of specified dimensions. Useful for initialization and placeholders in algorithms.

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ones

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Creates a matrix of specified size filled with ones. Often used for biases, pattern matrices, or simple tests.

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eye

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Generates the identity matrix $I$ with ones on the diagonal and zeros elsewhere. The identity acts as the multiplicative neutral element for matrix multiplication.

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rand / randn

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rand produces uniformly distributed random entries, while randn yields normally distributed entries with mean 0 and variance 1. Both are used for randomized tests, Monte Carlo, and initializing algorithms.