## 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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