Let 'I' be a linear transformation and let u₁, U2, U3} be a set of three vectors in the domain of T. • Statement (a) below is TRUE. I want you to prove it. • Statement (b) below is FALSE. I want you to give me an example of three specific vectors u, v, w and a specific linear transformation T that shows the statement is false. No further explanation is required. (a) If {u₁, U₂, U3} is linearly dependent, then {T(u₁), T(U₂), T(U3)} is linearly dependent. (b) If {U₁, U₂, U3} is linearly independent, then {T(U₁), T(U₂), T(13)} is linearly independent. To prove (a), I would start by writing down what it means for {u, v, w} to be a linearly dependent set according to the definition of "linearly dependent set." (When you don't know what else to do, invoke the definition; it gives you something specific to write down and work with.)
Let 'I' be a linear transformation and let u₁, U2, U3} be a set of three vectors in the domain of T. • Statement (a) below is TRUE. I want you to prove it. • Statement (b) below is FALSE. I want you to give me an example of three specific vectors u, v, w and a specific linear transformation T that shows the statement is false. No further explanation is required. (a) If {u₁, U₂, U3} is linearly dependent, then {T(u₁), T(U₂), T(U3)} is linearly dependent. (b) If {U₁, U₂, U3} is linearly independent, then {T(U₁), T(U₂), T(13)} is linearly independent. To prove (a), I would start by writing down what it means for {u, v, w} to be a linearly dependent set according to the definition of "linearly dependent set." (When you don't know what else to do, invoke the definition; it gives you something specific to write down and work with.)
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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Transcribed Image Text:Let T be a linear transformation and let {U₁, U₂, U3} be a set of three
vectors in the domain of T.
• Statement (a) below is TRUE. I want you to prove it.
• Statement (b) below is FALSE. I want you to give me an example of
three specific vectors u, v, w and a specific linear transformation
T that shows the statement is false. No further explanation is
required.
(a) If {u₁, U₂, U3} is linearly dependent, then
{T(u₁), T(u₂), T(13)} is linearly dependent.
(b) If {u₁, U₂, U3} is linearly independent, then
{T(u₁), T(u₂), T(13)} is linearly independent.
To prove (a), I would start by writing down what it means for {u, v, w}
to be a linearly dependent set according to the definition of "linearly
dependent set." (When you don't know what else to do, invoke the
definition; it gives you something specific to write down and work with.)
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