Let T: Rm → R" and S: RRP be linear transformations. Then So T: RRP is a linear transformation. Moreover, their standard matrices are related by[So T] = [S][T].Verify the result of the theorem above for the following S and T by finding the matrix of S o 7 by direct substitution and by matrix multiplication of [S][T].8-22-2-43X3(a) by direct substitution(b) by matrix multiplication

The objective of the question is to verify the theorem that states the composition of two linear…

Answered: Let T: Rm → R" and S: RRP be linear…

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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Write a basis for the Image of the transformation associated to x + 4y 2x + 8y X V The first vector of the basis has components 1 and and the second vector in the basis has components Write N for the last two entries if the second vector is not needed. and
4a). The transformation defined by the matrix A = [14 5 stretches images in R² in the directions y = x and y = -x. Figure out the factor by which anything in the y = x direction is stretched and the factor by which anything in the y = -x direction is stretched. 4b). The transformation defined by the matrix B = -82 -55 13 3] stretches images in R2 in one direction by a factor of 3 and some other direction by a factor of 2. Figure out what direction gets stretched by a factor of 3 and what direction gets stretched by a factor of 2. 4c). The transformation defined by the matrix C = stretches images in R2 in two directions. Find the directions and the factors by which it stretches in those directions.
Question 6 The matrix A is the standard matrix for the transformation x→ Ax that maps A² [79] O 8 -12 -3 5 J بر ده 31 — O2 3 71 O 8 -3 -12 5 into and maps into }] . Find
-9 V₁ = --[-] and X₂ such that T(x) is Ax for each x. Let x = A = X₁ -2 [3] and let T: R² R² be a linear transformation that maps x into x₁v₁ +X₂V₂. Find a matrix A -8 and V₂ =
Let T and T2 two linear transformations from R? to R², defined by 7(;) - ) ()-() 3x - 2y (;) (} x+ y T1 T2 y - 3x for all (x, y)" e R?. Find the standard matrix representations for each of the following composite functions, and their inverse transformation, if they exist, by using the algorithm for matrix inversion in Lemma 3, Topic 14. )T followed by T2. (i) T2 followed by an anticlockwise rotation by angle 5 in R?. (ii) A reflection in the line y = 2x followed by a projection onto the line y = -3x in R2.
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.
Covert 47 meter into kilometers
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7. Is there a linear transformation T : R³ → R³ such that T If so, what is its matrix?
. Linear transformation T: Rm →R" is given by one of the matrices below. Find m and n. 1 (a) [ 1 4 ] [ (b) 4 -2 3 8 i] 0 11 (c) 2-2 3 -3
Consider the linear transformation from R² to R² known as “"rotation by -90 de- grees." The matrix of this transformation is Is this transformation onto ? Is this transformation injective (one-to-one) ? What is the image of 0 1 [99] -1 0 1 2 ? If there is an inverse for this transformation, what is the matrix of the inverse?
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