Linear Algebra and Its Applications, Exercise 2.3.18

Exercise 2.3.18. Indicate whether the following statements are true or false:

a) given a matrix A whose columns are linearly independent, the system Ax = b has one and only solution for any right-hand side b

b) if A is a 5 by 7 matrix then the columns of A cannot be linearly independent

Answer: a) False. If A has fewer columns than rows then the system Ax = b has more equations than unknowns and may not have a solution. For example, if

A = \begin{bmatrix} 1&0 \\ 0&1 \\ 0&1 \end{bmatrix} \quad \rm and \quad b = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}

corresponding to the system

\setlength\arraycolsep{0.2em}\begin{array}{rcrcl}x_1&&&=&0 \\ &&x_2&=&0 \\ &&x_2&=&1 \end{array}

then the columns of A are linearly independent but the system Ax = b has no solution since the second and the third equations result in a contradiction.

b) True. If A is a 5 by 7 matrix then it has seven columns, each of which is an element of \mathbf{R}^5. But it is impossible to have more than five linearly independent vectors in \mathbf{R}^5 so the columns of A must be linearly dependent.

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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