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I am studying linear algebra opencourseware by gilbert strang.

This is the question posed. 

Construct a matrix whose column space contains (1, 1, 0) and
(0, 1, 1) and whose nullspace contains (1, 0, 1) and (0, 0, 1).

Solution 

Solution. (4 points) Not possible : Such a matrix A must be 3 × 3. Since the nullspace is supposed to
contain two independent vectors, A can have at most 3−2 = 1 pivots. Since the column space is supposed to
contain two independent vectors, A must have at least 2 pivots. These conditions cannot both be met!

The point i would like to check is. How did he come to the conclusion that the matrix must be a 3X3 matrix? Why cannot it be a 3 x 4 or any other matrix?

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    It's probably implicitly understood to be square. – 2017-01-06
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    @Thoth why is this the case? – 2017-01-06

1 Answers 1

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The nullspace is a subset of the domain, the column space of the codomain. Both of those are $\mathbb R^3$, so the matrix must be $3\times3$.