cos y cos asin y cos 3 sin a- sin y cos acos y cos sin asin 3 sin asin y sin 3cos y sin a + sin y cos 3 cos a- siny sin a + cos y cos cos a- sin 3 cos acos y sin 3cos 33. With the transformation matrix (A) given by (4.20), explicitly verify that3Σaijaik = δjk,i=1for (a) j = 1, k = 1, (b) j = 1, k = 2, (c) j = 1, k = 3.(A) =

Answered: Image /qna-images/answer/b165f5b2-c08c-4497-b5dd-3116c633d225.jpg

Answered: 3. With the transformation matrix (A)…

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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Suppose that T: R" → Rm is defined by T() = A for each of the following matrices below: [40] E = 0 0 02 0 D = 0 -1 0 0 0 0.5 [201] F = 10 3 01 (a) Rewrite T: R" → Rm with correct numbers for m and n for each transformation. What is the domain and codomain of each transformation? (b) Find some way to explain in words and/or graphically what each transformation does as it takes vectors from R to Rm. You might find it t helpful to try out a few input vectors and see what their image is under the transformation. This might be difficult, but an honest effort will give you credit. (c) For the transformation, can you get any output vector? (Any vector in R™) i. If so, explain why you can get any vector in Rm. ii. If not, give an example of an output vector you can't get with the transformation and explain why.
2. Let A = be an orthogonal 2 × 2 matrix. cos(0) sin(0 (a) Explain why the vector can be written as for some value of 0. a Use your answer to part (a) to find all 2×2 orthogonal matrices A = |C (b) (Hint: What can the second column of A look like?) (c) det(A) = 1? What kind of linear transformation do they correspond to? ! Given your answer to part (b), what are the 2 × 2 orthogonal matrices with
Find the matrix of the linear transformation T from R2 - R² where T([1, 0]) = [1, –2] and T([2, 1]) = [2, 3] %3D (Note: matrix A = [T([1, 0]), T([0, 1])] 1 -2 (a) 1 (b) 1 (c) -2 (d) -2 3 7 3 Ob a C
3 1 Let TA: R2 R³ be the matrix transformation corresponding to A = 1 -2 5 ТA(u) 3 18 -1 = TA(V) X -1 4 Find TA(u) and TA(V), where u = 1 3 [2] and v = [-2]
9. Consider the following 2 x 2 matrix A and basis S of R²: A ¹ = [² %) and 5 = {[_¹₂] · [_³;]} The matrix A defines a linear operator on R². Find a matrix B that represents the mapping A relative to the basis S.
Let [09] 6 1 Find the matrix C of the linear transformation T(x) = B(A(x)). C = A = and B = 24 0
Let L: R3 R2 be defined by the following. X1 + X3 ***] 8x2 Suppose X1 L X₂ P = = 1 -(HH) 4 3 3 1 2 B₁ = 4 ={[8][;}} is an ordered basis for the domain and B₂ = is an ordered basis for the range. Find the matrix representation P for L relative to B₁ and B₂ such that [L(u)]b₂ = P[u]ß₁'

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3. With the transformation matrix (A) given by (4.20), explicitly verify that 3 Σaijaik = δjk, i=1 for (a) j = 1, k = 1, (b) j = 1, k = 2, (c) j = 1, k = 3.