Question 629335:
Express the vector v= [-17] as a linear combination of x = [-2] and y = [-5]
[ 50 ] [ 5 ] [ -6]
v=__x + __ y.
How could I express this? Answer by jsmallt9(3758) (Show Source):
The problem is to find what number times x plus what number times y adds up to v. Let's call these two numbers a and b. So we're looking for what a and b make
v = a*x + b*y
true.
Substituting the vectors/column matrices in for v, x and y out equation is:
Performing the scalar multiplication on the right side we get:
Adding the vectors/column matrices on the right we get:
By the definition of matrix/vector equality, these are equal only if each element of one is equal to the corresponding element of the other. So
-17 = -2a + (-5b) and
50 = 5a + (-6b)
To find the correct values for a and b we solve this system. You have probably learned several ways to solve a system of linear equations like this: Substitution, Linear Combination/Elimination, several matrix methods, determinants/Cramer's Rule, etc. I'm going to use Linear Combination (but any of them will work and give the right answers).
Multiplying the first equation by 5 and the second equation by 2 we get:
-85 = -10a + (-25b)
100 = 10a + (-12b)
Adding the two equations together we get:
15 = -37b
Dividing both sides by -37 we get:
-15/37 = b
Now we use this value and one of the earlier equations to find a:
Simplifying we get:
Subtracting 75/37 to each side:
Dividing by -2:
(