## Linear Algebra and Its Applications, Exercise 2.4.1

Exercise 2.4.1. Suppose that for an $m$ by $n$ matrix $A$ we have $m = n$. State whether the following is true or false: The row space $\mathcal{R}(A^T)$ and column space $\mathcal{R}(A)$ of $A$ are the same.

Answer: Consider the following example of a 2 by 2 echelon matrix $U$ with a single pivot: $U = \begin{bmatrix} 1&2 \\ 0&0 \end{bmatrix}$

Since the second row is zero the first row is the only vector contributing to the row space $\mathcal{R}(U^T)$ and thus the vector $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ is a basis for $\mathcal{R}(U^T)$. In geometric terms the row space $\mathcal{R}(U^T)$ is the line represented by the equation $y = 2x$.

Since the only pivot is in the first column that column $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ is a basis for the column space $\mathcal{R}(U)$. In geometric terms the column space $\mathcal{R}(U)$ is the x-axis.

So the row space $\mathcal{R}(U^T)$ and column space $\mathcal{R}(U)$ are not equal for this example matrix $U$, even though the number of rows $m$ of $U$ is the same as the number of columns $n$. The statement above is therefore false.

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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