## Linear Algebra and Its Applications, Exercise 2.4.16

Exercise 2.4.16. Given an $m$ by $n$ matrix $A$ the columns of which are linearly independent, fill in the blanks in the following statements: The rank of $A$ is ____. The nullspace is ____. The row space is ____. There is at least one ____-inverse.

Answer: If the columns of $A$ are linearly independent then the rank of $A$ is $r = n$, the number of columns.

In the system $Ax = 0$ the product $Ax$ of $A$ and the vector $x = \begin{bmatrix} x_1&x_2&\dotsc&x_n \end{bmatrix}^T$ is a linear combination of the columns of $A$ with $x_1, x_2, \dotsc, x_n$ as coefficients. Since the columns of $A$ are linearly independent, the only way for the linear combination $Ax$ to be zero is for $x_1 = x_2 = \cdots = x_n = 0$. The nullspace $\mathcal N(A)$  is therefore the set containing only the zero vector.

Since the dimension of the row space $\mathcal R(A^T)$  is the same as the dimension of the column space $\mathcal R(A)$ , the dimension of the row space must be $n$. Each row has $n$ entries and is thus an element of $\mathbf R^n$ and since it has dimension $n$ the row space $\mathcal R(A^T)$  itself is equal to $\mathbf R^n$.

Since the columns of $A$ are linearly independent and the rank $r = n$ there is at least one left-inverse of $A$.

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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