Which mathematical concept describes the scalars by which a linear transformation stretches or compresses vectors along specific directions?

Which mathematical concept describes the scalars by which a linear transformation stretches or compresses vectors along specific directions?

• A. Eigenvalues
• B. Eigenvectors
• C. Determinants
• D. Trace

Answer: (A)

Scaling factors along directions that stay the same under a linear transformation are called eigenvalues. For a square matrix A, if there exists a nonzero vector x such that Ax = λx, then λ is the eigenvalue associated with that eigenvector x. This λ tells you how much the vector is stretched (λ > 1), compressed (0 < |λ| < 1), or flipped (λ negative) along that particular direction. If a vector is not an eigenvector, the transformation changes its direction, so that simple scaling doesn’t describe it. The other concepts describe different ideas: eigenvectors identify the invariant directions, the determinant measures overall volume scaling, and the trace is the sum of eigenvalues.

Scaling factors along directions that stay the same under a linear transformation are called eigenvalues. For a square matrix A, if there exists a nonzero vector x such that Ax = λx, then λ is the eigenvalue associated with that eigenvector x. This λ tells you how much the vector is stretched (λ > 1), compressed (0 < |λ| < 1), or flipped (λ negative) along that particular direction. If a vector is not an eigenvector, the transformation changes its direction, so that simple scaling doesn’t describe it. The other concepts describe different ideas: eigenvectors identify the invariant directions, the determinant measures overall volume scaling, and the trace is the sum of eigenvalues.