## 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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