MATH 235 W22 - A4
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MATH 235 W22 - A4
Question 1
For i = 1, 2, 3, 4, determine whether the function Fi is a linear transformation or not. If the function Fi is a linear transformation, then prove it.
If the function Fi is a not a linear transformation, then prove that it is not.
(a) F1 : P1 (R) → P1 (R), F1 (p(x)) = p(0) + p(1)x
(b) F2 : P1 (R) → P1 (R), F2 (p(x)) = p(0)p(1)x
(c) F3 : P2 (R) → P2 (R), F3 (p(x)) = p(0) + ╱ │x=–3 、x + ╱ │x=5、x2
(d) F4 : P1 (R) → M2乂2 (R), F4 (p(x)) = ╱ p(2) + p( −2) p( −2) 、
. (p(2))(p( −2)) p(2) − p( −2) .
Question 2
Let T : C → V be a linear transformation from C to the vector space V . Let c ∈ C, with c 0, satisfy T (c) = v, for some v ∈ V. What is T (x), x ∈ C?
Question 3
Let T : V → W be a linear transformation where V and W are two vector spaces. Suppose that {v1 , v2 , v3 } is a linearly dependent subset of V .
Show that {T (v1 ), T (v2 ), T (v3 )} is a linearly dependent subset of W .
Question 4
Let V, W, and X be vector spaces. Let T1 : V → W and T2 : W → X be linear transformations. Prove that the composite function T2 ◦ T1 : V → X, defined by (T2 ◦ T1 )(v) = T2 (T1 (v)), ∀v ∈ V, is a linear transformation.
Question 5
Let A, B ∈ Mn 乂n (F). Show that if A is similar to B, then rank(A) = rank(B), that is, prove that similar matrices have the same rank.
(Hint: show that A and B have the same nullity by relating their nullspaces, or make use of linear transformations.)
Question 6
Let T : V → W be a linear transformation where V and W are two vector spaces. Let U be a vector subspace of V.
Let the restriction of T to U, denoted by T |U : U → W, be the function defined by
T |U (u) = T (u), ∀u ∈ U.
Show that T |U is a linear transformation.
2022-02-16