linear algebra 2.2 Q11 11. a. Prove that if T: R" → R" is a linear transformation and c is any scalar, then thefunction cT: R" → R" defined by (cT)(x) = cT (x) (i.e., the scalar c times thevector T (x)) is also a linear transformation.b. Prove that if S: R" → R" and T : R"function S +T: R"→ R" are linear transformations, then the→ R" defined by (S + T)(x) = S(x) + T (x) is also a lineartransformation.c. Prove that if S: R"function SoT : R" → RP is also a linear transformation.→ RP andT: R"→ R" are linear transformations, then the

Answered: 11. a. Prove that if T : R" –→ R" is 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 2.2 Q11

11. a. Prove that if T: R" → R" is a linear transformation and c is any scalar, then the
function cT: R" → R" defined by (cT)(x) = cT (x) (i.e., the scalar c times the
vector T (x)) is also a linear transformation.
b. Prove that if S: R" → R" and T : R"
function S +T: R"
→ R" are linear transformations, then the
→ R" defined by (S + T)(x) = S(x) + T (x) is also a linear
transformation.
c. Prove that if S: R"
function SoT : R" → RP is also a linear transformation.
→ RP andT: R"
→ R" are linear transformations, then the

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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