defenition study

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Linear Independence

A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others.

Linearly Dependent

A set of vectors is linearly dependent if at least one of the vectors is a linear combination of the others.

Zero Vector in a Set

If a set of vectors includes the zero vector, it is automatically linearly dependent.

Orthogonal Vectors

Two vectors are orthogonal if their dot product is zero.

Orthonormal Vectors

A set of vectors is orthonormal if each vector has unit length and they are all mutually orthogonal.

Orthogonal Matrix

A square matrix is orthogonal if its columns (and rows) are orthonormal vectors, i.e., ( Q^T Q = I ).

Jordan Canonical Form

A form of a square matrix where it is broken into Jordan blocks corresponding to each eigenvalue, including generalized eigenvectors.

Diagonalizable Matrix

A matrix is diagonalizable if it has enough linearly independent eigenvectors to form a diagonal matrix under similarity transformation.

Similar Matrices

Two matrices are similar if they represent the same linear transformation under different bases, i.e., ( A = PBP^{-1} ).

Properties of Similar Matrices

Similar matrices have the same characteristic polynomial, eigenvalues, and determinant.

Singular Value Decomposition (SVD)

A = UΣVᵀ, where U and V are orthogonal matrices, and Σ contains the singular values on the diagonal.

Singular Values

The non-negative square roots of the eigenvalues of ( A^T A ); they measure how A stretches space.

U and V in SVD

The columns of U are eigenvectors of ( AA^T ); the columns of V are eigenvectors of ( A^T A ).

QR Factorization

A = QR, where Q is an orthogonal matrix and R is an upper triangular matrix.

Q in QR Factorization

Q contains orthonormal columns that span the column space of A.

R in QR Factorization

R is an upper triangular matrix that represents the projection coefficients from the Gram-Schmidt process.