linear algebra (a) Let L: R R be the linear transformation that projects a vector in R2 onto the r-axis. Let M be the 2 × 2matrix such that L() Ma.Give one eigenvector and associated eigenvalue for M. In this problem it is fine to give a thorough geometricexplanation without finding the matrix M.(b) Proposition 1 Let T: V W be a one-to-one linear transformation of linear spaces V and W, suppose that a setof vectors {1, Tg, U3} CV is lincarly independent. Prove that {T(71),T(52), T(3)} W is linearly independent.Below is a "proof" of the above proposition. In the spaces, fill in the missing parts to complete the proof or giveyour own proof.Proof: Suppose that a1T(,) + a2T(2) + a3T(73) = 0. Then, since T is a linear transformation we getT(-Since T is one-to-one, we get thataj01+ a202+ az03=%3DSince {1, 2, 03} is a linearly independent setwe get thatThus {T(1),T(2), T(U3)} CW is linearly independent.

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Answered: (b) Proposition 1 Let T: V W be a…

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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linear algebra

(a) Let L: R R be the linear transformation that projects a vector in R2 onto the r-axis. Let M be the 2 × 2
matrix such that L() Ma.
Give one eigenvector and associated eigenvalue for M. In this problem it is fine to give a thorough geometric
explanation without finding the matrix M.
(b) Proposition 1 Let T: V W be a one-to-one linear transformation of linear spaces V and W, suppose that a set
of vectors {1, Tg, U3} CV is lincarly independent. Prove that {T(71),T(52), T(3)} W is linearly independent.
Below is a "proof" of the above proposition. In the spaces, fill in the missing parts to complete the proof or give
your own proof.
Proof: Suppose that a1T(,) + a2T(2) + a3T(73) = 0. Then, since T is a linear transformation we get
T(-
Since T is one-to-one, we get that
aj01+ a202+ az03=
%3D
Since {1, 2, 03} is a linearly independent set
we get that
Thus {T(1),T(2), T(U3)} CW is linearly independent.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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