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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