Mathematics 244 – Linear Algebra Discussion 2 – “Compatible” Mappings February 18, 2004

Background

Today we will begin to consider the main topic of the second unit of the course, namely mappings T : V → W where V and W are both vector spaces. It will turn out that the most interesting mappings here are ones that are “compatible” in a suitable sense with the vector space structures (that is the vector sum and scalar multiplication operations) in V and W. Part of the goal of today’s discussion is to identify what this “compatibility” should mean.

Discussion Questions

A. Let V = W = R^{2}, and consider the mapping R_{θ} : R^{2} → R^{2} that takes each vector x to the vector R_{θ}(x) obtained by rotating by an angle of θ counterclockwise about the origin.

1) For instance, with θ = π/2, explain why R_{π/2}((1, 0)) = (0, 1) and R_{θ}((0, 1)) = (−1, 0). What is R_{π/2}((−1/2, \sqrt{3}/2))?

2) Show that for a general vector (x, y), R_{π/2}((x, y)) = (−y, x).

3) Suppose you apply R_{π/2} to a linear combination of two vectors and compute

R_{π/2}(c(x_{1}, y_{1}) + (x_{2}, y_{2})) = R_{π/2}((cx_{1} + x_{2}, cy_{1} + y_{2})).

How does the result relate to R_{π/2}((x_{1}, y_{1})) and R_{π/2}((x_{2}, y_{2}))?

4) More generally, the rotation mapping R_{θ} is defined by

R_{θ}((x, y)) = (cos(θ)x − sin(θ)y, sin(θ)x + cos(θ)y)

Suppose you apply R_{θ} to a linear combination of two vectors and compute

R_{θ}(c(x_{1}, y_{1}) + (x_{2}, y_{2})) = R_{θ}((cx_{1} + x_{2}, cy_{1} + y_{2})).

How does the result relate to R_{θ}((x_{1}, y_{1})) and R_{θ}((x_{2}, y_{2}))?

B. Let V = R^{3}, W = R^{2} and consider the mapping L : V → W defined by

L((x_{1}, x_{2}, x_{3})) = (x_{1} + 2x_{2}, 3x_{1} + 4x_{3}).

1) What are L((1, 0, 0)), L((0, 1, 0)) and L((0, 0, 1))?

2) Suppose you apply L to a linear combination of two vectors and compute

L((cx_{1} + y_{1}, cx_{2} + y_{2}, cx_{3} + y_{3})).

How does the result relate to L((x_{1}, x_{2}, x_{3})) and L((y_{1}, y_{2}, y_{3}))?

C. Now let V = C^{1}(R) (the space of all continuously differentiable functions – f ∈ V means f′ exists and is continuous on R). Let W = C(R). Consider the mapping D : V → W defined by D(f) = f′ (that is D takes a function f ∈ V to its derivative function f′ ∈ W).

1) What is D(5e^{7x} + cos(3x))? How does D(5e^{7x} + cos(3x)) relate to D(e^{7x}) and D(cos(3x))?

2) For general f, g ∈ V, how does D(cf + g) relate to D(f) and D(g)? Explain how this is similar to what you saw in A parts 3 and 4.

D. Let V = C(R) and W = R. Define I : V → W by I(f) = ∫_{0}^{1} f(x) dx (that is I takes the function f to its definite integral over the interval [0, 1]).

1) What is I(5e^{7x}+cos(3x))? How does I(5e^{7x}+cos(x^{2})) relate to I(e^{7x}) and I(cos(3x))?

2) For general f, g ∈ V, how does I(cf + g) relate to I(f) and I(g)? Explain how this is similar to what you saw in A parts 3 and 4, and B part 2.

E. Thinking about A part 4, and parts 2 of B, C and D, what is the corresponding general definition of a property of T : V → W? Explain why this property might appropriately be called a sort of “compatibility” with the vector space structures in V and W.

Assignment

Group write-ups due Monday, February 23.