a. Prove that if T : Rn Rm is a linear transformation and c is any scalar, then the function cT : Rn Rm 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 : Rn Rm and T : Rn Rm are linear transformations, then the function S + T : Rn Rm defined by (S + T )(x) = S(x) + T (x) is also a linear transformation. c. Prove that if S : Rm Rp and T : Rn Rm are linear transformations, then the function ST : Rn Rp is also a linear transformation.

Fall 2011 MA 16200 Study Guide - Exam # 1 ▯ (1) Distance formula D = (x2− x 1 + (y −2y ) 1 (z − 2 ) ;1equation of a sphere with cen- 2 2 2 2 ter (h,k,l) and radius r: (x − h) + (y − k) + (z − l) = r . −→ (2) Vectors in R and R ; displacement...