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3. Let S: U → V and T: V → W be linear transformations between finite dimen- sional vector spaces. (a) Let {v1, v2,.n} eV be a linearly independent set and {d1, A2, ..n} be scalars such that A #0 for all i. Show that the set of vectors {A1v1, Azv2,,.n U'n} are linearly independent. (b) Let v, we V. Show that span(v, w) = span(v, v + w). Hint: Proced in two steps: i. Show that is re span(v, w) implies ze span(v, v + w). ii. Show that is re span(v, v + w) implies re span(v, w). (c) Assume the rank of S = nullity of T. (range of S = null space of T). Assume S is 1-1 and T is onto. %3D Show that: dimV = dimU + dimW by using the rank-nullity theorem.

3. Let S: U → V and T: V → W be linear transformations between finite dimen- sional vector spaces. (a) Let {v1, v2,.n} eV be a linearly independent set and {d1, A2, ..n} be scalars such that A #0 for all i. Show that the set of vectors {A1v1, Azv2,,.n U'n} are linearly independent. (b) Let v, we V. Show that span(v, w) = span(v, v + w). Hint: Proced in two steps: i. Show that is re span(v, w) implies ze span(v, v + w). ii. Show that is re span(v, v + w) implies re span(v, w). (c) Assume the rank of S = nullity of T. (range of S = null space of T). Assume S is 1-1 and T is onto. %3D Show that: dimV = dimU + dimW by using the rank-nullity theorem.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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3. Let S: U → V and T : V → W be linear transformations between finite dimen- sional vector spaces. (a) Let {v1, v2,.n} e V be a linearly independent set and {A1, A2, ....n} be scalars such that A; # 0 for all i. Show that the set of vectors {A1v1, Azv2, , ....n Un} are linearly independent. (b) Let v, we V. Show that span(v, w) = span(v, v + w). Hint: Proced in two steps: i. Show that is a e span(v, w) implies re span(v, v + w). ii. Show that is re span(v, v + w) implies re span(v, w). (c) Assume the rank of S = nullity of T. (range of S = null space of T). Assume S is 1-1 and T is onto. Show that: dimV = dimU + dimW by using the rank-nullity theorem.
Transcribed Image Text:3. Let S: U → V and T : V → W be linear transformations between finite dimen- sional vector spaces. (a) Let {v1, v2,.n} e V be a linearly independent set and {A1, A2, ....n} be scalars such that A; # 0 for all i. Show that the set of vectors {A1v1, Azv2, , ....n Un} are linearly independent. (b) Let v, we V. Show that span(v, w) = span(v, v + w). Hint: Proced in two steps: i. Show that is a e span(v, w) implies re span(v, v + w). ii. Show that is re span(v, v + w) implies re span(v, w). (c) Assume the rank of S = nullity of T. (range of S = null space of T). Assume S is 1-1 and T is onto. Show that: dimV = dimU + dimW by using the rank-nullity theorem.
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