MAT A22
Homework # 5 –
Winter 2024
Homework Guidelines
This homework was released on Fri. Feb. 9th 14:00 (EST). It is due on Fri. Feb. 16th 17:00 (EST).
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For this assignment, Question 3 must be typed up in LATEX.
Readings
❼ ➜2.1 Linear Transformations
❼ ➜2.2 Linear Transformations between Finite-Dimensional Vec-tor Spaces
Problems
Q1. Let L(V, W) be the set of linear transformations from V to W. Prove that the set L(V, W) forms a vector space using the following operators. Let S, T ∈ L(V, W) and c ∈ F
(S ⊞ T)(v) = S(v) ⊕ T(v)
(c ⊡ T)(v) = c ⊙ T(v)
where ⊕ is the vector addition operator from W and ⊙ is the scalar multiplications operators from W. (Remember to show closed un-der addition and multiplication.) Suppose, V and W are both finite dimensional, determine the dimension of L(V, W).
Q2. For each of the following, determine if T : V → W is a linear transformation. Provide an appropriate proof if true and a counter example if false.
(a) V = R2 and W = R 2 . T(x1, x2) = (x1 − x2, 3x2 + 2x1)
(b) V = R2 and W = R 2 . T(x1, x2) = (x1x2, x2 − x1)
(c) V = R2 and W = R 2 . T(x1, x2) = (1 − x1, x2)
(d) V = R2 and W = R 3 . T(x1, x2) = (0, x1 − x2, x1 + x2)
Q3. (******This question is to be typed up in LATEX******)
Let V = R 3 and W = R 2 . Let T ∈ L(V, W) such that
T(1, 1, 0) = (2, 1), T(1, 0, 1) = (1, 1) and T(0, 1, 1) = (−1, 2).
Let v1 = (1, 1, 0), v2 = (1, 0, 1) and v3 = (0, 1, 1).
(a) Express x ∈ V as a linear combination of v1, v2, v3.
(b) Compute T(x).
Q4. Let V = R 3 and W = R 2 . Let T ∈ L(V, W) such that
T(1, 1, 1) = (2, 1), T(1, 2, 1) = (−1, 1) and T(3, 1, 1) = (−1, 2).
Let v1 = (1, 1, 1), v2 = (1, 2, 1) and v3 = (3, 1, 1).
(a) Express x ∈ V as a linear combination of v1, v2, v3.
(b) Compute T(x).
In questions 5 and 6 you will explore the standard inner product a little more. Recall the definition of the standard inner product.
Q5. Prove that the standard inner product satisfies the following prop-erties.
(a) ⟨⃗u, ⃗v⟩ = ⟨⃗v, ⃗u⟩
(b) ⟨⃗u, ⃗v1 + ⃗v2⟩ = ⟨⃗u, ⃗v1⟩ + ⟨⃗u, ⃗v2⟩
(c) ⟨⃗u, a⃗v⟩ = a⟨⃗v, ⃗u⟩
(d) ⟨⃗u, ⃗u⟩ ⩾ 0, with equality if ⃗u = 0.
Q6. Let V = R n . Suppose W is a subset of V . The set W⊥ is the set of vectors perpendicular to W.
W⊥ = {x ∈ R n : ⟨x, v⟩ = 0, for all v ∈ W}
Prove that W⊥ is a subspace of V . (You may reference Q5a, Q5b, Q5c, or Q5d as justification if needed)
Q7. Consider a = (x1, x2, x3) ∈ R3 , b = (y1, y2, y3) ∈ R3 . The cross product of a and b, denoted a×b, is given by the following equation
a × b = (x2y3 − x3y2, x3y1 − x1y3, x1y2 − x2y1)
Prove the following:
(a) a × b ∈ (span({a, b}))⊥
(b) If a and b are collinear, then a × b = 0.
(c) What is the dimension of (span({a, b}))⊥ if
1. a and b are collinear?
2. a and b are not collinear?
Q8. Let V be the set of continuous real valued functions. Define the appropriate vector space W for each situation and prove that the following well know functions are linear transformations.
(a) The derivative operator.
(b) The definite integration operator Int(f) = 
f(x)dx.
(c) The definite integration operator Int(f) = 
f(t)dt.