Consider a transformation T: R³-R4. Given the following sets U and V of 4 vectors in the domain and codomain of T, respectively, |> ul, u2, u3, u4 := (1, 0, 1), (2, 1, 0), (0, 2, 1), (1, 4, 2); v1, v2, v3, v4 := (-2, 0, 1, 1), (1, 1, 0, 1), (-2, 0, 0, 0), (2,2,-3,-1); T (uk) = Vk² such that T =v, for k= 1, 2, 3, 4: ul, u2, u3, u4 := vl, v2, v3, v4 := HO 0 (a) Identify a subset of vectors {u} that make a basis B in R³ (explain). Express the remaining vector u in U as a linear combination of the basis vectors, and show that the linearity property of T is satisfied, i.e. that the image T(`u__) of the linearly dependent vector u can be produced in the codomain by the same linear combination of images of the B-basis vectors, {v=T(uk) }· (b) Find now the image T(u5) of the vector u5 = = (0, 1,-3); 4-8 u5 := -3
Consider a transformation T: R³-R4. Given the following sets U and V of 4 vectors in the domain and codomain of T, respectively, |> ul, u2, u3, u4 := (1, 0, 1), (2, 1, 0), (0, 2, 1), (1, 4, 2); v1, v2, v3, v4 := (-2, 0, 1, 1), (1, 1, 0, 1), (-2, 0, 0, 0), (2,2,-3,-1); T (uk) = Vk² such that T =v, for k= 1, 2, 3, 4: ul, u2, u3, u4 := vl, v2, v3, v4 := HO 0 (a) Identify a subset of vectors {u} that make a basis B in R³ (explain). Express the remaining vector u in U as a linear combination of the basis vectors, and show that the linearity property of T is satisfied, i.e. that the image T(`u__) of the linearly dependent vector u can be produced in the codomain by the same linear combination of images of the B-basis vectors, {v=T(uk) }· (b) Find now the image T(u5) of the vector u5 = = (0, 1,-3); 4-8 u5 := -3
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:Consider a transformation T: R³ R4. Given the following sets U and V of 4 vectors in the domain and
codomain of T, respectively,
|> ul, u2, u3, u4 := (1, 0,— 1), (2, 1, 0), (0, 2,— 1), (1, 4, 2);
v1, v2, v3, v4 := (–2, 0, 1, 1), (− 1, 1, 0, 1), (−2, 0, 0, 0), (2, 2,-3,-1);
(b) Find now the image T(u5) of the vector
[> u5 :=
ul, u2, u3, u4 :=
(0, 1,-3);
vl, v2, v3, v4 :=
such that T(uk) = for k= 1, 2, 3, 4 :
(a) Identify a subset of vectors {u} that make a basis B in R³ (explain).
Express the remaining vector u in U as a linear combination of the basis vectors, and show that the
linearity property of T is satisfied, i.e. that the image T(`u___`) of the linearly dependent vector u can be produced
in the codomain by the same linear combination of images of the B-basis vectors, {vk=T(uk)}.
BAB:
0
0
--B
u5 :=
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