Concept: The problem requires determining the coefficients of a linear combination of the standard basis vectors, which is achieved by solving a system of linear equations derived from matrix multiplication.
Formula/Principle: Matrix multiplication: If
M is a matrix and
v is a vector, the resulting vector is found by multiplying the matrix by the vector.
Solution/Analysis:1.
Determine the Matrix M: Since the columns of a matrix
M are the results of multiplying
M by the standard basis vectors
(),
(), and
(), we can construct
M:
M=()2.
Set up the System of Equations: We are given that
M()=(). Performing the matrix multiplication yields the following system of linear equations:
1x+0y−1z=1(1)2x+1y+1z=7(2)3x+2y+1z=11(3)3.
Solve the System:- From equation (1), we get: x−z=1, which implies x=1+z.
- Substitute x into equation (2):
2(1+z)+y+z=7
2+2z+y+z=7
y+3z=5(4) - Substitute x into equation (3):
3(1+z)+2y+z=11
3+3z+2y+z=11
2y+4z=8
Dividing by 2, we get: y+2z=4(5)
4.
Solve for y and z: Subtract equation (5) from equation (4):
(y+3z)−(y+2z)=5−4z=1Substitute
z=1 back into equation (5):
y+2(1)=4y=25.
Solve for x: Use the relation
x=1+z:
x=1+1x=26.
Calculate the Final Value: The question asks for the value of
x+y+z:
x+y+z=2+2+1=5