Question

Theorem 1.8.2 T: $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is a matrix transformation if and only if the following relationships hold for all vectors u and v in $\mathbb{R}^n$ and for every scalar k: (i) T(u + v) = T(u) + T(v) [Additivity property] (ii) T(ku) = kT(u) [Homogeneity property]

          Theorem 1.8.2

T: $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is a matrix transformation if and only if the following relationships hold
for all vectors u and v in $\mathbb{R}^n$ and for every scalar k:
(i) T(u + v) = T(u) + T(v) 		[Additivity property]
(ii) T(ku) = kT(u) 		[Homogeneity property]

Theorem 1.8.2

T: ℝ^n →ℝ^m is a matrix transformation if and only if the following relationships hold
for all vectors u and v in ℝ^n and for every scalar k:
(i) T(u + v) = T(u) + T(v) [Additivity property]
(ii) T(ku) = kT(u) [Homogeneity property]

Added by Ronald M.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Michelle Z.

03/15/2023

Step 1

Step 1: The theorem states that a function T from R^n to R^m is a matrix transformation if and only if it satisfies two properties: additivity and homogeneity. Show more…

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Could you please explain these laws? I want to understand without simply regurgitating the information. Thank you!! Theorem 1.8.2: T: R^(n) -> R^(m) is a matrix transformation if and only if the following relationships hold for all vectors u and v in R^(n) and for every scalar k: (i) T(u+v) = T(u) + T(v) [Additivity property] (ii) T(ku) = kT(u) [Homogeneity property] Theorem 1.8.2: T: R^n -> R^m is a matrix transformation if and only if the following relationships hold for all vectors u and v in R and for every scalar k: (i) T(u+v) = T(u) + T(v) [Additivity property] (ii) T(ku) = kT(u) [Homogeneity property]