Exercise 1.1.9

Question

Suppose the column space of an $m$ by $n$ matrix is all of $\mathbf{R}^{3}$. What can you say about $m$? What can you say about $n$ ? What can you say about the rank $r$?

Neil Z. · Accepted Answer

If the column space space of an $m$ by $n$ matrix is all of $R^3$, then the dimension of column space is 3, i.e. the rank $r=3$. Also because the dimension of the column space is less or equal to the number of columns, we have $m\ge 3$. Also we know that column rank = row rank, so we also have $n\ge 3$.

Exercise 1.1.9

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Exercise 1.1.9

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