Consider linear transformations mapping vectors in R2 to vectors in R, clearly these are maps from...

  

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Consider linear transformations mapping vectors in R2 to vectors in R, clearly these are maps from a 2D vectors space to a 1D one. Since the space mapped to is one dimensional, we can call the mappings linear functions. Let e₁,e₂ be a basis for R². Clearly a linear function on f: R² → R can be defined for all v E R² by giving its values for the basis vectors f(e₁), f(e₂). E.g. if f(e₁) = 2, f(e₂) = 5, then the vector 6e₁7e₂ is mapped: f(6e₁ - 7e₂) = 6f (e₁)-7 f(e₂) = 12 - 35 = -23. Using this property, let us define the two linear functions h₁, h₂, by h₁ (e₁) = 1, h₁(e₂) = 0; h₂ (e₁) = 0,h₂ (e₂) = 1. For any linear function g: R² → R and any vector v E R², g(v) = g(e₁)h₁(v) + g(e₂)h₂ (v). (i) Prove that this last equation holds, (ii) Let the basis for R² be e₁ = (2), e₂ = (3) If your vectors are not independent, then adapt them. A linear function will be represented by a 1 × 2 matrix. Give matrices for: h₁, h₂ in the standard basis and a matrix for the linear function defined by 6 (e₁) = 5,6 (e₂) = 6, again in the standard basis. Prove that h₁, h₂ form a basis for the space of all linear functions using the result proved in (i) or otherwise. (iii) Consider linear transformations mapping vectors in R2 to vectors in R, clearly these are maps from a 2D vectors space to a 1D one. Since the space mapped to is one dimensional, we can call the mappings linear functions. Let e₁,e₂ be a basis for R². Clearly a linear function on f: R² → R can be defined for all v E R² by giving its values for the basis vectors f(e₁), f(e₂). E.g. if f(e₁) = 2, f(e₂) = 5, then the vector 6e₁7e₂ is mapped: f(6e₁ - 7e₂) = 6f (e₁)-7 f(e₂) = 12 - 35 = -23. Using this property, let us define the two linear functions h₁, h₂, by h₁ (e₁) = 1, h₁(e₂) = 0; h₂ (e₁) = 0,h₂ (e₂) = 1. For any linear function g: R² → R and any vector v E R², g(v) = g(e₁)h₁(v) + g(e₂)h₂ (v). (i) Prove that this last equation holds, (ii) Let the basis for R² be e₁ = (2), e₂ = (3) If your vectors are not independent, then adapt them. A linear function will be represented by a 1 × 2 matrix. Give matrices for: h₁, h₂ in the standard basis and a matrix for the linear function defined by 6 (e₁) = 5,6 (e₂) = 6, again in the standard basis. Prove that h₁, h₂ form a basis for the space of all linear functions using the result proved in (i) or otherwise. (iii)

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