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Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution? Discuss.

it is not possible.

Calculus 3

Chapter 4

Vector Spaces

Section 6

Rank

Vectors

Harvey Mudd College

Baylor University

University of Nottingham

Boston College

Lectures

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for a problem problem 22. So where we need to figure out what it is possible for it is possible that also shows up a homogeneous system up a 10 linear equations with 12 variables are multiples off one fixed, none zero solution. So, actually, this is equivalent to say we have a matrix that is 10 by 12 and we have the phone so much in his system, X equals zero. So this is equivalent to say it's impossible for a for A to have dimension open all surveys one. Because if we if every every solutions in this system can be expressed as a multiple upper uppers, upper multiple off one's fixed and that's your solution, that means every solution can be express as this form. So we must have ah, re variable. We don't We don't know which one it is, but it must. There must exist a free sample so that every variable in this factor will be determined by this variable. So so she can be expressed as a multiple wall boat about non zero, a fixed and then zero social. So Okay, so what can we see to this extent? So, um, we first by wreck fear. Um, I wrecked your, um We know that rank a fe us. You mentioned nothing. All space. Okay, ISS 12. So we suppose suppose you mentioned on the north face is one this one, Then the rank will be nothing. So is it possible for rank to rack up A to B 11? Well, it's none. It's not possible because the rank of a should be smaller or equal to 10. But Frank okay, should be smaller. Way to 10. So we have a contradiction. So that means it's impossible two.

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