10. Let L: R R' be a map defined as follows:(4x1 +4 x2 - 2 x3 -9 x45x1 +2 x2 - 3 x3-9 x46x) – 4 x3 - 9 x47x1-2x2-5x3-9 x4)X2X3X4,(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 obtainedin part (c) matches the statement of the theorem.

Given : L :ℝ4→ℝ4 be a map defined as…

Answered: 10. Let L: R'- R' be a map defined as…

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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Similar questions

1). Corresponds to the Transformation k(x,y) = ( x + 3, y-5). Fully describe the transformation K. 2). Corresponds to the Transformation H(x,y) = (0.5x, 0.5y). Fully describe the transformation H.
B) IR2 --> IR3 (x, y) --> (x, y2, x+y) Is it linear transformation? investigate.
Suppose T : R^2 → R^2 is a linear transformation that does not alter e_i andwhich maps e_2 into e_2 + 2e_1, where e_i = (1,0), e_2 = (0,1). Find the transformation matrix T.Justify your answer.
3. (1 point) Let f : R² → R' be the linear transformation determined by -2 -2 (:) -5 3 -3 4 -6 -9- (1) Find f -7 (5)-|-} f (2) Find the matrix of the linear transformation f. X y y (3) The linear transformation f is • injective
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.
2 -2 -[:]~-~-[;) - - [³ ]-~^»-[*³] and and e₂= and Y₂ = 5 6 Let e, = into y₂. Find the images of The image of 5 [3] -3 is 5 5] -3 and X₂ X2 and let T. R² R² be a linear transformation that maps e, into y, and maps e₂ →
Let f: R R be defined by f((x, y)) = 7y - 3x - 3. Is f a linear transformation? a. f((x₁, y₁) + (x₂, y₂)) = etc.) f((x₁, y₁>) + f((x₂, y₂)) = + Does f((x1, y₁) + (x2, Y2 )) = f ((x₁, y₁ )) + f((x2, Y₂ )) for all (x₁, y₁), (x2, Y₂) E R ²? choose b. f (c(x, y)) = c(f((x, y))) = Does f(c(x, y)) = c(f ((x, y))) for all c ER and all (x, y) ER ²? choose c. Is f a linear transformation? choose . (Enter x₁ as x1, <
8-3) show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,... are not vectors but are entries in vectors a) T(x1, x2) = (2x2-3x1, x1-4x2, 0, x2) b) T(x1, x2, x3, x4) = 2x1 + 3x - 4x4 (T:R* → R)
Suppose T: M2,2→→P2 is a linear transformation whose action on a basis for M2,2 is as follows: [18]-- Describe the action of T on a general matrix, using x as the variable for the polynomial and a, b, c, and d as constants. Use the character to indicate an exponent, e.g. ax^2-bx+c. = -2x+5 T = 0 1 1 -2 -2 = -x+4 T -4 -2 = 2x² +4x-26 T 00 = -2x²+2x+10
QI) Write down the transformation T (B} 2, 2, (4} 2

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