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10. Let L: R'- R' be a map defined as follows: (4x1+4x-2x-9 x4 5x1 +2x2 -3 x)-9x4 6x -4 x - 9 x4 (7x)-2x2-5x3-9x4) L. (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.

10. Let L: R'- R' be a map defined as follows: (4x1+4x-2x-9 x4 5x1 +2x2 -3 x)-9x4 6x -4 x - 9 x4 (7x)-2x2-5x3-9x4) L. (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.

Advanced Engineering Mathematics
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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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.
Transcribed Image Text: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.
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