Question

Suppose you are in the garden supply business, offering three fertilizer brands: Vigoro, Parker’s, and Bleyer’s. The nutrient content per 100 pounds for each brand is represented by the nutrient vectors:

v = [18; 65; 198]
p = [4; 13; 39]
b = [3; 12; 37]
​
Determine the linear transformation R³ -> R³ that takes a vector x of brand amounts (in hundreds of pounds) as input and produces the nutrient vector as output:
t(x) = [ ... ; ... ; ... ] x Then, find a formula for t⁻¹ and use it to determine the amounts of Vigoro, Parker’s, and Bleyer’s required to produce 139 pounds of nitrogen, 502 pounds of phosphoric acid, and 1529 pounds of potash:
t⁻¹([139; 502; 1529]) = [ ... ; ... ; ... ]

A)t(x) = [18x₁ + 4x₂ + 3x₃; 65x₁ + 13x₂ + 12x₃; 198x₁ + 39x₂ + 37x₃]
B)t(x) = [4x₁ + 13x₂ + 12x₃; 18x₁ + 65x₂ + 198x₃; 3x₁ + 39x₂ + 37x₃]
C)t(x) = [3x₁ + 12x₂ + 37x₃; 4x₁ + 13x₂ + 39x₃; 18x₁ + 65x₂ + 198x₃]
D)t(x) = [37x₁ + 12x₂ + 3x₃; 39x₁ + 13x₂ + 4x₃; 198x₁ + 65x₂ + 18x₃]
E)t(x) = [3x₁ + 4x₂ + 18x₃; 12x₁ + 13x₂ + 65x₃; 37x₁ + 39x₂ + 198x₃]
​

Answer 1

The linear transformation that takes a vector x of brand amounts as input and produces the nutrient vector as output is given by t(x) = [18x₁ + 4x₂ + 3x₃; 65x₁ + 13x₂ + 12x₃; 198x₁ + 39x₂ + 37x₃]. The formula for the inverse transformation, t⁻¹, is [37x₁ + 12x₂ + 3x₃; 39x₁ + 13x₂ + 4x₃; 198x₁ + 65x₂ + 18x₃].

Step-by-step explanation:

The linear transformation that takes a vector x of brand amounts (in hundreds of pounds) as input and produces the nutrient vector as output is given by:

t(x) = [18x₁ + 4x₂ + 3x₃; 65x₁ + 13x₂ + 12x₃; 198x₁ + 39x₂ + 37x₃]

The formula for t⁻¹, the inverse transformation, can be found by solving the equation t(x) = [139; 502; 1529:

t⁻¹([139; 502; 1529]) = [37x₁ + 12x₂ + 3x₃; 39x₁ + 13x₂ + 4x₃; 198x₁ + 65x₂ + 18x₃]