Y Let V = {p(x) E P2(R) | p(-2) = 0} and W = E R³ | y– 2w +z = 0 be vector spaces. Consider the map T : V → W defined by*) a T(a(-2 – a) + b(4 – a*) = a, b E R. b. -a + 26 ) Show that T satisfies the additive property of a linear transformation (map). ) Show that V is isomorphic to W (i.e., V and W are isomorphic). Remark: If you wish: B = {p1(x), p2(x)} = {-2 – x, 4 – x²} is a basis for V, and is a basis for W.
Y Let V = {p(x) E P2(R) | p(-2) = 0} and W = E R³ | y– 2w +z = 0 be vector spaces. Consider the map T : V → W defined by*) a T(a(-2 – a) + b(4 – a*) = a, b E R. b. -a + 26 ) Show that T satisfies the additive property of a linear transformation (map). ) Show that V is isomorphic to W (i.e., V and W are isomorphic). Remark: If you wish: B = {p1(x), p2(x)} = {-2 – x, 4 – x²} is a basis for V, and is a basis for W.
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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Transcribed Image Text:4. Let V = {p(x) E P2(R) | p(-2) = 0} and W
E R | y – 2w+z = 0 } be vector spaces.
Consider the map T : V → W defined by(*)
a
T(a(-2 – x) + b(4 – x²))
a, b e R.
—а + 2b
(A) Show that T satisfies the additive property of a linear transformation (map).
(B) Show that V is isomorphic to W (i.e., V and W are isomorphic).
(*) Remark: If you wish: B = {P1(x), p2(x)} = {-2 – x, 4 – x²} is a basis for V,
and
C =
is a basis for W.
2
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