## Linear Algebra and Its Applications, Review Exercise 2.28

Review exercise 2.28. a) If $A$ is an $m$ by $n$ matrix with linearly independent rows, what is the rank of $A$? The column space of $A$? The left null space of $A$?

b) If $A$ is an 8 by 10 matrix and the nullspace of $A$ has dimension 2, show that the system $Ax = b$ has a solution for any $b$.

Answer: a) If $A$ has $m$ rows and they are linearly independent then the rank of $A$ is $r = m$.

Since the rank is $m$ there are also $m$ linearly independent columns of $A$ and the column space is $\mathbb{R}^m$. (In other words, the $m$ linearly independent columns span $\mathbb{R}^m$.)

The dimension of the left nullspace is then $m-r = m-m = 0$. The left nullspace thus contains only the zero vector.

b) Since the dimension of the nullspace of a matrix is $n-r$ in general, if the dimension of the nullspace of $A$ is 2 we have $2 = 10-r$ or $r = 8$. Since the rank of $A$ is equal to the number of rows of $A$, by 20Q on page 96 there exists a solution to the system $Ax = b$ for every $b$.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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