Exercise 2.4.12. Suppose that for a matrix the system has at least one nonzero solution. Show that there exists at least one vector for which the system has no solution. Show an example of such a matrix and vector .
Answer: Suppose is an by matrix. If there exists a nonzero for which then the columns of are linearly dependent. (If they were linearly independent then would imply that for all .) We therefore have the rank .
Since and the dimension of the row space is equal to the dimension of is less than . In other words, the rows of do not span all of .
But if that is the case then there exists at least one vector in that is not in and cannot be expressed as a linear combination of the rows of . Therefore for the vector there is no solution to the system . (If there were such a solution then its coefficients would define a linear combination of the rows of equal to .)
For example, suppose that we have the following 2 by 2 matrix
The system is equivalent to
This system has rank with as a basic variable and as a free variable.
From the first equation of the system we have or . If we set the free variable to 1 we then have and as a (nonzero) solution to .
The vector is a basis for the row space , which consists of all vectors of the form . Geometically this is a line through the origin and the point .
Pick a point in the x-y plane not on this line, for example , and consider the vector and the system
The first equation of the system produces while the second equation produces , a contradiction. There is thus no solution to for the example values of and .
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.