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Problem 8 Let V = R², W₁ = Sp{(0, 1)}, and W₂ = Sp{(1, 1)}. a. Prove that V = W₁ © W₂. b. Find a formula for T₁(x, y), where T₁ is defined as in the previous problem.

Problem 8 Let V = R², W₁ = Sp{(0, 1)}, and W₂ = Sp{(1, 1)}. a. Prove that V = W₁ © W₂. b. Find a formula for T₁(x, y), where T₁ is defined as in the previous problem.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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solve problem 8 base on problem 7

Problem 7 W₂ and proved that any x Є V can Recall that on homework 2, we defined the direct sum V = W₁ be uniquely written as x = x₁ + x2, with x₁ € W₁ and 22 € W₂. Using this notation, the map T₁(x) = x₁ is called the projection of V on W₁ (this is well-defined since ₁ is unique). We can similarly define T₂(x) = x2, the projection of V on W2. Prove that T₁ is a linear transformation. Prove also that W₁ = {x € V | T₁(x) = x}.
Transcribed Image Text:Problem 7 W₂ and proved that any x Є V can Recall that on homework 2, we defined the direct sum V = W₁ be uniquely written as x = x₁ + x2, with x₁ € W₁ and 22 € W₂. Using this notation, the map T₁(x) = x₁ is called the projection of V on W₁ (this is well-defined since ₁ is unique). We can similarly define T₂(x) = x2, the projection of V on W2. Prove that T₁ is a linear transformation. Prove also that W₁ = {x € V | T₁(x) = x}.
Problem 8 Let V = R², W₁ = Sp{(0, 1)}, and W₂ = Sp{(1, 1)}. a. Prove that V = W₁ W₂. b. Find a formula for T₁(x, y), where T₁ is defined as in the previous problem.
Transcribed Image Text:Problem 8 Let V = R², W₁ = Sp{(0, 1)}, and W₂ = Sp{(1, 1)}. a. Prove that V = W₁ W₂. b. Find a formula for T₁(x, y), where T₁ is defined as in the previous problem.
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