7. Let T : R4 → R* be a linear transformation whose standard matrix is74106.16-4128127-6-2Determine if T is a one-to-one mapping.8. Let T : R5 → R be a linear transformation whose standard matrix is9.136.1415-7-6-8-912-5-9-5-6-89.8131415211Determine if T is an onto mapping.

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Answered: 7. Let T: R4 → R4 be a 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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5. Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x₁, x2, and x3 are not vectors but are entries of a vector x in R³. T(x1, x2, x3) = (2x1 — x2, x1 − 2x2 + x3, x1 − 3x2 + x3) - X1
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.
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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)
A linear transformation T : R³ → R² whose matrix is -2 -6 -6 -2 -6 -10 + k| is onto if and only if k +

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7. Let T: R4 → R4 be a linear transformation whose standard matrix is 7 4 10 6. 16 -4 12 12 -8 -6 -2 Determine if T is a one-to-one mapping.