Let B be the set of vectors in R^3 given by
B = (1, 1, 0) , (1, 1, 1) , (0, 1, −1).
1. For x = (x1, x2, x3)^T a vector in R^3, find the scalars c1, c2, c3 (that will
depend on x1, x2, x3) such that
x = c1(1, 1, 1) + c2(1, 1, 1) + c3(0, 1, −1)
Can anyone help me solve this? Showing the steps to solve this would be great.
Thank You.
To solve this question, we need to find the values of c1, c2, and c3 such that the given vector x can be written as a linear combination of the vectors in set B.
Step 1: Express x as a linear combination of the vectors in set B.
x = c1(1, 1, 1) + c2(1, 1, 1) + c3(0, 1, −1)
Step 2: Expand the expression.
x = (c1 + c2)(1, 1, 1) + c3(0, 1, −1)
Step 3: Equate the components of the vectors.
x1 = c1 + c2
x2 = c1 + c2 + c3
x3 = c1 - c3
Step 4: Solve the system of equations for c1, c2, and c3.
We can solve this system of equations by substitution or elimination method.
From equation 1, we have c1 = x1 - c2. Substitute this into equation 2 to get:
x2 = (x1 - c2) + c2 + c3
x2 = x1 + c3
Now we can solve for c3:
c3 = x2 - x1
Substitute the value of c3 into equation 3 to get:
x3 = (x1 - c3) - c3
x3 = x1 - 2c3
Now we can solve for c1:
c1 = x1 - c2
c1 = x1 - (x2 - x1)
c1 = 2x1 - x2
Finally, we can substitute the values of c1 and c3 into equation 1 to solve for c2:
x1 = c1 + c2
x1 = (2x1 - x2) + c2
c2 = x2 - x1
So, the scalars c1, c2, and c3 are given by:
c1 = 2x1 - x2
c2 = x2 - x1
c3 = x2 - x1
Therefore, the scalars c1, c2, and c3 are dependent on the values of x1, x2, and x3.