Determine (without doing row operations) that a system is consistent

[JJBladester]

Hello,

I just started learning Elementary Linear Algebra. I've read through the first chapter in my book (Elementary Linear Algebra by Larson, Edwards, Falvo, 5th edition). There were two "Discovery" questions that try to see if you know how systems of equations work. I am unsure on both of them.

These are not homework questions, rather a test to see if you know how to determine if a system is consistent or if a system has an infinite # of solutions.

Consider the system of linear equations.
2x₁ + 3x₂ + 5x₃ = 0
-5x₁ + 6x₂ -17x₃ = 0
7x₁ - 4x₂ + 3x₃ = 0

Without doing any row operations, explain why this system is consistent.

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The following system has more variables than equations. Why does it have an infinite number of solutions?
2x₁ + 3x₂ + 5x₃ + 2x₄ = 0
-5x₁ + 6x₂ -17x₃ - 3x₄ = 0
7x₁ - 4x₂ + 3x₃ + 13x₄ = 0

Since I am just beginning to learn linear algebra, I am not sure exactly the answer for either question. I know a system is consistent if it has exactly one solution or if it has infinite solutions, so how could I tell if the system for question one is consistent without doing any row operations?

For the second problem, I know that a system has infinite solutions if after doing row operations you get a free variable, but how do you determine if a variable is considered free?

 

[JJBladester]

Ok... I read further along in the book and now I understand that for the first question, any homogeneous system will have at least one solution (where all variables = 0). Still wondering about question two.

 

[JJBladester]

Ok... I found the solution to the second question. Every homogeneous system of linear equations is consistent. Moreover, if the system has fewer equations than variables, then it must have an infinite number of solutions. Bingo. Next time I'll read a bit further before posting.

 

[PowerIso]

Well, glad you answered your own question. Keep these ideas in mind though, they'll come in handy once you reach vector spaces.

 

[JJBladester]

Thanks for the encouragement and the heads-up, PowerIso. I'll make sure to bank this knowledge and look for it to come up again soon. I found some online Linear Algebra videos from MIT here:

http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/index.htm

Between my school's lectures and these, I am becoming more matrix-confident.