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Chapter 4: Q10E (page 191)

If the null space of A \({\bf{7}} \times {\bf{6}}\) matrix A is 4-dimensional, what is the dimension of the column space of A?

Short Answer

Expert verified

The dimension of the column space of A is 1.

Step by step solution

01

Find the rank of the matrix

Using the nullity theorem, you get:

\(\begin{aligned} {\rm{rank}}\,A &= n - \dim \,{\rm{Nul}}\,A\\ &= 6 - 5\\ &= 1\end{aligned}\)

02

Find the dimension of the column space

The dimension of the column space of A is equal to its rank.

So, the dimension of the column space of A is 1.

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Most popular questions from this chapter

Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent? Explain.

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Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

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