Homework Help is Here – Start Your Trial Now!
Math
Algebra
Problem 4. Recall that in the notes, we thought about the following transformation: Let V be an n-dimensional vector space and let S:= {d1, 02, ..., n} be any basis for V. Let vE V, and suppose that = c01 + c202 + ...+ Cnn. So define T :V R" by T(7):= (c1, C2,..., Cn). (a) Prove that T is well defined; that is, prove that each vector in V has exactly one image under T. This means proving two things: (1) that every vector in V has an image under T, and (2) that each vector in V has only one image under T. (b) Prove that T is a linear transformation. (c) Prove that T is one to one. (d) Prove that T is onto.

Problem 4. Recall that in the notes, we thought about the following transformation: Let V be an n-dimensional vector space and let S:= {d1, 02, ..., n} be any basis for V. Let vE V, and suppose that = c01 + c202 + ...+ Cnn. So define T :V R" by T(7):= (c1, C2,..., Cn). (a) Prove that T is well defined; that is, prove that each vector in V has exactly one image under T. This means proving two things: (1) that every vector in V has an image under T, and (2) that each vector in V has only one image under T. (b) Prove that T is a linear transformation. (c) Prove that T is one to one. (d) Prove that T is onto.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 18EQ
See similar textbooks
icon
Related questions
Question

help on part b

Problem 4. Recall that in the notes, we thought about the following transformation: Let V be an n-dimensional vector space and let S :={v1, 02, . . . , Tn} be any basis for V. Let EV, and suppose that ở = c101 + c22 + + + Cn Un. So define T: V → R" by %3D T(T) := (c1, c2, ..., Cn). (C1, C2, .. . (a) Prove that T is well defined; that is, prove that each vector in V has exactly one image under T. This means proving two things: (1) that every vector in V has an image under T, and (2) that each vector in V has only one image under T. (b) Prove that T is a linear transformation. (c) Prove that T is one to one. (d) Prove that T is onto. 1
Transcribed Image Text:Problem 4. Recall that in the notes, we thought about the following transformation: Let V be an n-dimensional vector space and let S :={v1, 02, . . . , Tn} be any basis for V. Let EV, and suppose that ở = c101 + c22 + + + Cn Un. So define T: V → R" by %3D T(T) := (c1, c2, ..., Cn). (C1, C2, .. . (a) Prove that T is well defined; that is, prove that each vector in V has exactly one image under T. This means proving two things: (1) that every vector in V has an image under T, and (2) that each vector in V has only one image under T. (b) Prove that T is a linear transformation. (c) Prove that T is one to one. (d) Prove that T is onto. 1
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS