Which of the following statements are true? Select all true statements! Only select true statements! (a) T :Rª → R defined by T((r", y)) = ry is a lincar transformation. (b) T : P3 → R defined by T(p) = p(4) is a lincar transformation. (c) If T : R" → R" is a l-1 map, then T is an isomorphism. (d) The vector spaces R? and P, are isomorphic. (c) A 3 x 3 matrix must have exactly 3 cigenvalues.

Which of the following statements are true? Select all true statements! Only select true statements! (a) T :Rª → R defined by T((r", y)) = ry is a lincar transformation. (b) T : P3 → R defined by T(p) = p(4) is a lincar transformation. (c) If T : R" → R" is a l-1 map, then T is an isomorphism. (d) The vector spaces R? and P, are isomorphic. (c) A 3 x 3 matrix must have exactly 3 cigenvalues.

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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3. Let T : R - R³ be a lincar transformation such that T(1, 1, 1) T(0,-1, 2) T(1,0, 1) = (1, 1,0). (2,0, –1) (-3,2, -1) %3D %3D %3D Find T(2, -1, 1).
Which of the following properties are true for the transformation T : R³ → R² with (E)- -y x + y (a) T is one to one (b) T is onto (c) T is linear (b) and (c) (a), (b), and (c) (a) and (b) (c) only OO O
Determine which of the following transformations are linear transformations. A. The transformation T defined by T(1, 2, X3) = (x1,0, X3) B. The transformation T defined by T(x1, x2, 3) = (x1, x2, -3) C. The transformation T defined by T(1, 2, 3) = (1, 2, 3) D. The transformation T defined by T(x₁, x2) = (2x1 - 3x2, x1 +4,5x₂). E. The transformation T defined by T(x₁, x2) = (4x₁ − 2x2, 3|x₂|). -
Q. 4 (a). Determine whether the set is orthogonal and then orthonormal 1 -1 ,0) 1 1 五 -2 ),(- %3B (b). Determine whether the mapping T:M,→R is a linear transformation, where, a =a² +b?
Show that TR4 → R4 given by T(w, x, y, z) = (z, -y, x, -w)) is an isomorphism. (Please find the definition of isomorphism in the notes before attempting this problem! And don't forget that you have to prove it is a linear transformation)
Let T : M2x2 (R) → M2x2 (R) be the linear transformation defined by "(: :)-( ) "). 2x z y - z+ w T 2w 2w – z Is T diagonalizable? If so, find B so that [T], is diagonal.
Prove that the given transformation is a linear transformation, using the definition. = Definition: A transformation T: R"→ R" is called a linear transformation if the following is true. 1. T(u + v) = T(u) + T(v) for all u and v in Rº 2. T(cv) = cT(v) for all v in R" and all scalars c -y x + 7y 8x - 5y Use the following remark. Remark: The definition of a linear transformation can be streamlined by combining (1) and (2) as shown below. T(C₁₂V₁ + C₂V₂) = C₁ T(v₁) + c₂ T(v₂) for all v₁, v₂ in R" and scalars C₁, C₂ T is a linear transformation if and only if T(C₁ V₁ + C₂ V₂) = C₂ T(v₁) + C₂ T(v₂), where v₁ = - [K] ([*]+[C]) T(C₂v₁ + C₂v₂) Then we get the following. 11 -(5₂81 +5₂Y₂) 41*1 (C₁x₂ + ₂x₂) + ( )(C₂x₂ + C₂Y₂) ]) ] )(c₁×₂ + €₂×₂) + (¯¯)(C₂Y/₂ + C₂Y/₂) -C1Y₁ ]) ₁₂x₁₂ + 1 +(C 14₁/₁ -Y₁ 1)9₁8/₁ V₂ +5₂₂ = [3]. -52/2 1) <₂*₂ + ( [ -1/₂ 1) ²₂/₂ ])/₂ X
When possible, an example of a linear transformation T that satisfies the condition in hint on each item
10. Let L: R R' be a map defined as follows: (4x1 +4 x2 - 2 x3 -9 x4 5x1 +2 x2 - 3 x3-9 x4 6x) – 4 x3 - 9 x4 7x1-2x2-5x3-9 x4) X2 X3 X4, (a) Show that L is a linear transformation on R'. (b) Write out the matrix S which represents L with respect to the standard basis. (c) Find a basis for Range S and Ker S. (d) State the "rank-nullity theorem" and verify explicitly that the result obtained in part (c) matches the statement of the theorem.
If T : R3 [a] → M2x2 (R) is a linear transformation then T is an isomorphism. True O False