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Let $M$ be a $n\times m$ real matrix. Consider the following:

  • Let $k_1$ be the smallest number such that $M$ can be factorized as $A.B$, where $A$ is an $n\times k_1$ matrix and $B$ is a $k_1\times m$ matrix.
  • Let $k_2$ be the smallest number such that $M = \sum_{i=1}^{k_2}u_iv_i$, where $u_i$ is an $n\times1$ matrix and each $v_i$ is an $1\times m$ matrix.
  • Let $k_3$ be the column-rank of M.

Which of the following statements is TRUE?

(A) $k_1 < k_2 < k_3$

(B) $k_1 < k_3 < k_2$

(C) $k_2 = k_3 < k_1$

(D) $k_1 = k_2 = k_3$

(E) No general relationship exists among $k_1$, $k_2$ and $k_3$

in Linear Algebra by (3.6k points) | 5 views

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