Linear Combinations question

I just don't understand what they're asking for/what a combination is...

Explain why the system

u+ v+ w=2

u+2v+3w=1

v+2w=0

is singular by finding a combination of the three equations that adds up to 0=1. What value should replace the 0 on the right of the 3rd equation to allow the system to have solutions...what is one solution?

Write the system in a column picture. Show that the three columns on the left lie in the same plane by expressing the third column as a combination of the first two. WHat are all the solutions (u,v,w) if the column on the right side of the equation is changed to (0,0,0)?

Re: Linear Combinations question

Quote:

Originally Posted by

**jahichuanna** I just don't understand what they're asking for/what a combination is...

Explain why the system

u+ v+ w=2

u+2v+3w=1

v+2w=0

is singular by finding a combination of the three equations that adds up to 0=1. What value should replace the 0 on the right of the 3rd equation to allow the system to have solutions...what is one solution?

A "linear combination" of objects x, y, z (vectors, equations, or anything you can add and multiply by numbers) is ax+ by+ cy where a, b, and c are numbers. You are asked to find three numbers a, b, and c, such that a(u+ v+ w)+ b(u+ 2v+ 3w)+ c(v+ 2w)= 0. Of course, you would do the same on the right and have 0= 2a+ b. The problem is telling you that you should find that 2a+ b= 1. If you were to replace that 0 with x, then you would have 2a+ b+ cx. What should x be so 2a+ b+ cx= 0?

Quote:

Write the system in a column picture. Show that the three columns on the left lie in the same plane by expressing the third column as a combination of the first two. WHat are all the solutions (u,v,w) if the column on the right side of the equation is changed to (0,0,0)?

Again, a linear combination is simply sums of numbers times the objects, in this case the columns of the matrix. That is, if the matrix is

$\displaystyle \begin{bmatrix}a & b & c \\ d & e & f\\ g & h & i\end{bmatrix}$

then a "linear combination" of the columns are

$\displaystyle x\begin{bmatrix}a \\ d \\ g\end{bmatrix}+ y\begin{bmatrix}b \\ e \\ h\end{bmatrix}+ z\begin{bmatrix}c \\ f \\ i\end{bmatrix}$

for number x, y, and z.

I am concerned about your apparent ignorance of basic definitions. Whoever gave you these exercises clearly expects you to know what a "linear combination" is!

Re: Linear Combinations question

I realized I knew what it was...I just didn't know it by that name...I'm not sure why I hadn't heard of it, but I'll be sure to start reading up on my definitions! Thanks for your help! I got it all done :)