## Linear Algebra and Its Applications, Exercise 2.4.9

Exercise 2.4.9. Let $A$ be a matrix representing a system of $m$ equations in $n$ unknowns, and assume that the only solution to $Ax = 0$ is 0. What is the rank of $A$? Explain your answer.

Answer: $Ax$ is a linear combination of the columns of $A$ with the coefficients being the entries of $x$, namely $x_1, x_2, \dotsc, x_n$. If the only time when $Ax = 0$ is when $x$ itself is zero (i.e., the coefficients $x_1, \dotsc, x_n$ are all zero) then all the columns of $A$ are linearly independent. Since the rank $r$ of $A$ is the same as the number of linearly independent columns of $A$ we then have $r = n$.

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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