THEOREM 5.6 bon Proof If T V W and S: V→ W are linear transformations, then so are T + S and cT where c is a scalar. We verify this for T+S and leave the proof for cT as an exercise (Exercise 15). Suppose that u and u are vectors in V and a is a scalar. We have and (T+S) (u + v) = = T(u + v) + S(u + v) = T(u) + T(v) + S(u) + S(v) T(u) + S(u) + T (v) + S(v) = (T+S)(u) +(T+S)(v) (T+S) (av) = T(av) + S(av) = aT (v) +aS(v) = a(T(v) + S(v)) = a(T + S)(v) giving us T + S is a linear transformation. As an immediate consequence of Theorem 5.6, we get that linear combinations of linear transformations from one vector space to another are linear transformations.
THEOREM 5.6 bon Proof If T V W and S: V→ W are linear transformations, then so are T + S and cT where c is a scalar. We verify this for T+S and leave the proof for cT as an exercise (Exercise 15). Suppose that u and u are vectors in V and a is a scalar. We have and (T+S) (u + v) = = T(u + v) + S(u + v) = T(u) + T(v) + S(u) + S(v) T(u) + S(u) + T (v) + S(v) = (T+S)(u) +(T+S)(v) (T+S) (av) = T(av) + S(av) = aT (v) +aS(v) = a(T(v) + S(v)) = a(T + S)(v) giving us T + S is a linear transformation. As an immediate consequence of Theorem 5.6, we get that linear combinations of linear transformations from one vector space to another are linear transformations.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please write the solution on paper.
![**Theorem 5.6**
If \( T : V \to W \) and \( S : V \to W \) are linear transformations, then so are \( T + S \) and \( cT \) where \( c \) is a scalar.
**Proof**
We verify this for \( T + S \) and leave the proof for \( cT \) as an exercise (Exercise 15). Suppose that \( u \) and \( v \) are vectors in \( V \) and \( a \) is a scalar. We have
\[
(T + S)(u + v) = T(u + v) + S(u + v) = T(u) + T(v) + S(u) + S(v)
\]
\[
= T(u) + S(u) + T(v) + S(v) = (T + S)(u) + (T + S)(v)
\]
and
\[
(T + S)(av) = T(av) + S(av) = aT(v) + aS(v) = a(T(v) + S(v)) = a(T + S)(v)
\]
giving us \( T + S \) is a linear transformation.
As an immediate consequence of Theorem 5.6, we get that linear combinations of linear transformations from one vector space to another are linear transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfa44710-6742-4cf8-9a38-b63fd4325c9b%2F000d3a8b-d867-44a3-9605-f956f56971af%2Fitkpqbd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Theorem 5.6**
If \( T : V \to W \) and \( S : V \to W \) are linear transformations, then so are \( T + S \) and \( cT \) where \( c \) is a scalar.
**Proof**
We verify this for \( T + S \) and leave the proof for \( cT \) as an exercise (Exercise 15). Suppose that \( u \) and \( v \) are vectors in \( V \) and \( a \) is a scalar. We have
\[
(T + S)(u + v) = T(u + v) + S(u + v) = T(u) + T(v) + S(u) + S(v)
\]
\[
= T(u) + S(u) + T(v) + S(v) = (T + S)(u) + (T + S)(v)
\]
and
\[
(T + S)(av) = T(av) + S(av) = aT(v) + aS(v) = a(T(v) + S(v)) = a(T + S)(v)
\]
giving us \( T + S \) is a linear transformation.
As an immediate consequence of Theorem 5.6, we get that linear combinations of linear transformations from one vector space to another are linear transformations.
Transcribed Image Text:15. Complete the proof of Theorem 5.6 by showing that if \( T : V \rightarrow W \) is a linear transformation and \( c \) is a scalar, then \( cT \) is a linear transformation.
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
