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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