About: Linear Algebra Essentials

A matrix is a rectangular array of numbers. Its transpose \(A^T\) is obtained by swapping rows and columns, so \((A^T)_{ij} = A_{ji}\). The inverse \(A^{-1}\) of a square matrix is the unique matrix satisfying \(AA^{-1} = A^{-1}A = I\), where the identity matrix \(I\) has 1s on the diagonal and 0s elsewhere. Inversion...

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The determinant is a scalar that captures key information about a square matrix. For the 2×2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) it is simply \(ad - bc\). Geometrically, the determinant measures how the associated linear map scales area (in two dimensions) or volume (in higher dimensions); a zero de...

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An eigenvalue of a square matrix \(A\) is a scalar \(\lambda\) for which \(Av = \lambda v\) for some nonzero vector \(v\); the corresponding vector \(v\) is the eigenvector. Eigenvalues encode how \(A\) acts along preferred directions and are precisely the roots of the characteristic polynomial \(p(\lambda) = \det(A -...

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The inner (dot) product of two real vectors \(u\) and \(v\) is \(u \cdot v = \sum_i u_i v_i = u^T v\); it equals zero precisely when \(u\) and \(v\) are orthogonal. From it comes the Euclidean norm \(\|x\| = \sqrt{x \cdot x} = \sqrt{\sum_i x_i^2}\). The Cauchy–Schwarz inequality \(|u \cdot v| \leq \|u\| \|v\|\) follows...

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Factorizations express complicated matrices as products of simpler ones, exposing structure and enabling efficient computation. The LU decomposition writes a square matrix as \(A = LU\) with \(L\) lower triangular and \(U\) upper triangular, useful for solving linear systems by sequential substitution. It exists withou...

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For systems of linear equations, row operations — swapping two rows, multiplying a row by a nonzero scalar, or adding a multiple of one row to another — transform a matrix without changing the solution set of \(Ax = b\). Repeated application produces row echelon form, where each leading entry of a row lies to the right...

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