Chapter 4 - Vector Spaces - 4.9 The Rank-Nullity Theorem - Problems - Page 330: 14

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$A$ is a $3 \times 7$ matrix, $n=7$ According to Rank- Nullity Theorem, we obtain: $rank (A)+nullity (A)=n \\ rank (A)=n-nullity (A)=7-4=3$ But matrix $A$ has 3 rows, so $colspace (A) \in R^3$ The only subspace of $R^3$ is $R^3$. Hence, $colspace (A) = R^3$ Since $rowspace (A)$ is a subspace of $R^5$, it is not possible to say that $rowspace (A)=R^3$