## Linear Algebra and Its Applications, Exercise 2.3.21

Exercise 2.3.21. Suppose $A$ is a 64 by 17 matrix and has rank 11. How many independent vectors are solutions to the system $Ax = 0$? What about the system $A^Ty = 0$?

Answer: If the rank of $A$ is 11 then performing elimination on $A$ produces an matrix $U$ with 11 pivots and thus 11 basic variables. Since $U$ (like $A$) has 17 columns this means that there are 17 – 11 or 6 free variables that can be set to arbitrary values in solving the system $Ax = 0$. The nullspace of $A$ (i.e., the set of all vectors satisfying $Ax = 0$) therefore has dimension 6, and any basis for the nullspace has 6 linearly independent vectors each of which satisfy $Ax = 0$.

Since $A$ is 64 by 17 the matrix $A^T$ is 17 by 64. The original matrix $A$ had 11 pivots and 11 linearly independent rows. The rows of $A$ become columns in $A^T$ and thus $A^T$ has 11 linearly independent columns and also has rank 11. Since $A^T$ has 64 columns there are 64 – 11 or 53 free variables when considering the system $A^Ty = 0$. The nullspace of $A^T$ (i.e., the set of all vectors satisfying $A^Ty = 0$) therefore has dimension 53, and any basis for the nullspace has 53 linearly independent vectors each of which satisfy $A^Ty = 0$.

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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