Tutorial Video: Linear Combination & System of Equations
Example: Linear Combination of Vectors in R³
Let a₁ = [1, −2, 0], a₂ = [0, 1, 2], a₃ = [1, −2, 0] and b = [2, −1, 6].
Check whether b is a linear combination of a₁, a₂ and a₃.
Solution
x a₁ + y a₂ + z a₃ = b
Since a₁ = a₃, let t = x + z.
t[1,-2,0] + y[0,1,2] = [2,-1,6]
Comparing components:
t = 2
−2t + y = −1
2y = 6
−2t + y = −1
2y = 6
t = 2 and y = 3
2(3)=6 ✓
Since t = x + z = 2, infinitely many values of x and z exist. One choice is:
x = 2, y = 3, z = 0
b = 2a₁ + 3a₂ + 0a₃
✅ Therefore, b is a linear combination of a₁, a₂ and a₃.
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