Linear Systems: Linear combinations

In summary, the conversation discusses determining if b is a linear combination of a1, a2, and a3, using an augmented matrix and solving for the constants c1, c2, and c3. It is found that b is a linear combination of a1 and a2 when c3 is set to 0, but since c3 is a free variable, there are multiple linear combinations that result in b. The conversation also mentions the geometric interpretation of a linear combination and suggests verifying the answers and checking for errors in the book.
  • #1
Novean
4
0

Homework Statement



Determine if b is a linear combination of a1, a2, and a3.

a1= [1,-2, 0]
a2 = [0, 1, 2]
a3 = [5, -6 8]

b= [2, -1, 6]


The Attempt at a Solution



Alright, well I used an augmented matrix to solve the problem, and after reducing it completely, the matrix looked like this:

| 1 0 5 |2|
| 0 1 4 |3|
| 0 0 0 |0|

Solving for the c's, I got:
(Where the number to the right of the c denote which one instead of a multiplication number)
c1+ 5c3=2
c2+4c3=3
0c3=0

I found that b is a combination of a1 and a2 if c3 is set to 0, however, since c3 is a free variable, and therefore can be anything, then b wouldn't be a lin. combination if c3 be anything other than 0.

I'm having a hard time understanding this. Is it not a linear combination because there are other possible answers that wouldn't make it so? Or is it a linear combination because atleast one possibility works.

Thank you for any help.
 
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  • #2
Novean said:

Homework Statement



Determine if b is a linear combination of a1, a2, and a3.

a1= [1,-2, 0]
a2 = [0, 1, 2]
a3 = [5, -6 8]

b= [2, -1, 6]


The Attempt at a Solution



Alright, well I used an augmented matrix to solve the problem, and after reducing it completely, the matrix looked like this:

| 1 0 5 |2|
| 0 1 4 |3|
| 0 0 0 |0|

Solving for the c's, I got:
(Where the number to the right of the c denote which one instead of a multiplication number)
c1+ 5c3=2
c2+4c3=3
0c3=0

I found that b is a combination of a1 and a2 if c3 is set to 0, however, since c3 is a free variable, and therefore can be anything, then b wouldn't be a lin. combination if c3 be anything other than 0.
No, this isn't true. c3 can be set to any value. Of course, this will give different values for c1 and c2. The point is that there are lots of linear combinations of a1, a2, and a3 that produce b.

One thing to notice is that if c3 = 0, then c1 = 2 and c2 = 3. What this says is that b = 2a1 + 3a2.

Geometrically, b lies in the same plane as a1 and a2.
Novean said:
I'm having a hard time understanding this. Is it not a linear combination because there are other possible answers that wouldn't make it so? Or is it a linear combination because atleast one possibility works.

Thank you for any help.
 
  • #3
Ah! I understand.

Thanks, I forgot to also change the values for c1 and c2 when I picked another free c3.

What had me confused is that the book says it's not a combination. The answer on the back clearly states it in bold. However,the book is probably wrong (Which now a-days I find they often are in regards to answers in the back)
 
  • #4
First, make sure you are working the right problem.
Second, if you're working the same problem as in the book, check your answers. In this case, set c3 to some value (0 is convenient here) and verify that 2a1 + 3a2 actually adds up to b.

Try a different linear combination, by setting c3 to some other value (and calculating values for the two other constants). See if that linear combination of a1, a2, and a3 also ends up with b.
 

What is a linear system?

A linear system is a set of equations that can be represented using linear equations. These equations consist of variables and coefficients, and can be solved algebraically to determine the values of the variables.

What is a linear combination?

A linear combination is a mathematical operation that combines a set of vectors using scalar multiplication and vector addition. This allows us to create new vectors from existing ones, and is an important concept in linear algebra.

How do you determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. This means that each vector in the set is necessary to represent the entire set of vectors, and cannot be removed without changing the set.

What is the difference between a homogeneous and non-homogeneous linear system?

A homogeneous linear system is one where the constant terms in each equation are equal to zero. This means that the system has a unique solution or is inconsistent. A non-homogeneous linear system has non-zero constant terms, and can have either a unique solution, infinitely many solutions, or no solution at all.

How do you solve a linear system using Gaussian elimination?

Gaussian elimination is a method for solving linear systems by using elementary row operations to transform the system into an equivalent system with a simpler form. This process involves reducing the system to row-echelon form, and then back-substituting to find the values of the variables.

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