Same intrinsic rank, different packaging

Rank is intrinsic: you can embed $\mathbb{R}^2$ into $\mathbb{R}^{100}$ with a tall skinny matrix while keeping rank $2$ . Different shapes package the same linear information in different ambient coordinates.

Numerical rank uses singular values: tiny $\sigma_i$ signal approximate dependence. SVD writes $A=U\Sigma V^T$ with diagonal nonnegative $\Sigma$, separating energy along orthogonal input and output directions.

Left-multiplying by a full column-rank matrix changes column space unless the left factor is square invertible on the relevant span. Shape change can hide rank; singular values expose it. Two different-shaped matrices can both have rank $2$ while living in different ambient spaces; threshold tiny singular values to decide numerical rank when data is noisy. Column rank equals row rank even when $m\neq n$ . Embedding $\mathbb{R}^2$ into a high-dimensional space keeps rank $2$ while changing only packaging. SVD diagonal $\Sigma$ lists singular values that decay when noise is present. Threshold tiny $\sigma_i$ to read numerical rank. Intrinsic rank does not depend on embedding dimension. Compare SVD singular values before trusting raw matrix shape.