SOLUTION: Let u = (1,-1,3,5) and v = (2,1,0,-3). Find scalars a and b so that au + bv = (1,-4,9,18). I would like to know the steps on how to solve for scalars a and b.

Question 183071This question is from textbook Contemporary Linear Algebra
: Let u = (1,-1,3,5) and v = (2,1,0,-3). Find scalars a and b so that
au + bv = (1,-4,9,18).
I would like to know the steps on how to solve for scalars a and b. This question is from textbook Contemporary Linear Algebra

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
au+bv = (1,-4,9,18) ... Start with the given equation

a(1,-1,3,5)+b(2,1,0,-3) = (1,-4,9,18) ... Plug in the given vectors

(a,-a,3a,5a)+(2b,b,0,-3b) = (1,-4,9,18) ... Multiply the scalars by EVERY element in the vectors

(a+2b,-a+b,3a,5a-3b) = (1,-4,9,18) ... Add the vectors by adding the corresponding components.

Now because both vectors on both sides are equal in dimension, this tells us that the left components correspond to the right components. This means that , , , and

Which gives us the system of equations:

We can use any method to solve this system (substitution, elimination, matrix, etc), but notice how the third equation is composed of only one variable in which we can easily solve for.

Let's now solve the third equation

Start with the third equation.

Divide both sides by 3 to isolate "a".

Reduce.

-----------------------------------

Go back to the first equation.

Plug in

Subtract from both sides.

Combine like terms on the right side.

Divide both sides by to isolate .

Reduce.

========================================================

Answer:

So the scalars are and

In other words,

3(1,-1,3,5)-1*(2,1,0,-3) = (1,-4,9,18)
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