4. Let u₁ = (1,0, 0, 0), u2 = (1, 1, 0, 0), u3 = (1,1,1,0), u4 = (1, 1, 1, 1)be the vectors in R4. Suppose that TR4 → R6 is a mappingsuch that T(u₁) = T(u2) = T(U3) = T(u4) = 0 and T(4, 0, -2, 1) =(1, 0, 1,-1,2, 2). Determine whether T is a linear transformation.

Answered: Image /qna-images/answer/8a8eb7fc-4c1d-4664-bffb-7319ad6a9e1f.jpg

Answered: 4. Let u₁ (1,0, 0, 0), u2 = (1,1,0,0),…

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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18. Let V = {(a1, a2): a1, a2 € R}. For (a₁, a2), (b₁,b₂2) € V and c € R, define (a1, a2) + (b1,b2) = (a1 + 2b1, a2 + 3b2) and c(a1, a2) = (ca₁, ca₂). Is V a vector space over R with these operations? Justify your answer.
6. Let x#0 and y‡0. (a) If x ¹ y, show that {x, y} is a linearly independ- ent set. (b) Extend the result to mutually orthogonal nonzero vectors X1, Xm 2
29. Let T be linear transformation from a vector space U (F) into a vector space r) and a, a2, ..., C, e then show that a,, a2, E U. If T(a), T (a), . . , T (a,) are linearly independent ..., An are linearly independent.
4] Show that the following sets of vectors are linearly dependent in R3. (c) {{5, 2, -3), (3, 0, 4), (-3, 0, -4)} (d) {{1, 1, 1), (2, 2, 2), (0, 1, 5)}
4. Let T: U→V be a linear transformation from a vector space U (F) into a vector space V(F) and U(F) be finite dimensional. Show that U and R(T) have the same dimension iff T is nonsingular.
Let x = (1, 1, 1, 1)T and y = (8, 2, 2, 0)T . Find the vector projection p of x onto y
1- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (-3,0,4), (5,-1,2), (1,1,3) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 2- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthonormal basis for R³, then obtain an orthonormal basis from them. Express (3,7,-4) as a linear combination of the orthonormal basis. 3- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (1,1,0), (0,0,1), (0,1,1) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 4- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthogonal basis for R³, then obtain an orthonormal basis from them. Express (2,3,-4) as a linear combination of the orthonormal basis. 5- Find the coordinates of the vector u = (0,25, 1) with respect to the orthogonal basis v₁ = ( 3,4,0), V₂ = (−4, 3, 0 ) and v₂ = (−4,3,0) 6- Find the coordinates of the vector U = (7,5,2) with respect to the basis V₁ =…
Suppose T: R¹ → R™ is a linear transformation and {V₁, V2, V3}) is a set of vectors in R". (a) Complete the following definition: The set {V1, V2, V3} is called linearly independent if (b) (c) T is one-to-one if and only if Ker (T) = Show that if T is one-to-one and {V₁, V2, V3} is linearly independent, then the set {T(V₁), T(V₂), T(V3)} is linearly independent.

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4. Let u₁ (1,0, 0, 0), u2 = (1,1,0,0), u3 = (1,1,1,0), u4 = (1,1, 1, 1) be the vectors in R4. Suppose that TR4 R is a mapping such that T(u₁) = T(u₂) = T(u3) = T(u4) = 0 and T(4, 0, -2, 1) = (1,0, 1,-1, 2, 2). Determine whether T is a linear transformation.