A data scientist is compressing a 4-dimensional feature vector x by applying a linear change of coordinates. She collects four candidate vectors b₁, b₂, b₃, and b₄ ∈ ℝ⁴ and stacks them as the columns of a 4 × 4 matrix B, then checks that det(B) ≠ 0 before using B and B⁻¹ in her code. What does this determinant test actually guarantee about the set {b₁,…,b₄}?
They are eigenvectors of B, which means B must be symmetric.
They form a basis for ℝ⁴, allowing every feature vector to be expressed uniquely in their coordinates.
They minimize the variance of the transformed features, ensuring the transformation acts as a whitening matrix.
They are orthonormal, so multiplying by B preserves Euclidean distances.
A non-zero determinant implies that B is invertible. Invertibility of a square matrix is equivalent to its columns being linearly independent and spanning the entire space, so the four vectors constitute a basis for ℝ⁴ and provide a unique coordinate representation for every original feature vector. Orthogonality (distance preservation), being eigenvectors of B, or minimizing variance are separate properties that are not implied by det(B) ≠ 0.
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