Transcription of Matrix Representations of Linear Transformations and ...
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Matrix Representations of Linear Transformations and ...
Matrix Representations of Linear Transformations andChanges of subspaces and DefinitionsAsubspaceVofRnis a subset ofRnthat contains the zero element and is closed under additionand scalar multiplication:(1)0 V(2)u,v V= u+v V(3)u Vandk R= ku VEquivalently,Vis a subspace ifau+bv Vfor alla,b Randu,v V. (You should try to provethat this is an equivalent statement to the first.)Example {(t,3t, 2t)|t R}. ThenVis a subspace ofR3:(1)0 Vbecause we can taket= 0.(2)Ifu,v V, thenu= (s,3s, 2s)andv= (t,3t, 2t)for some real numberssandt. But thenu+v= (s+t,3s+ 3t, 2s 2t) = (s+t,3(s+t), 2(s+t)) = (t ,3t , 2t ) Vwheret =s+t R.
A subspace V of Rnis a subset of Rnthat contains the zero element and is closed under addition and scalar multiplication: (1) 0 2V (2) u;v 2V =)u+ v 2V (3) u 2V and k2R =)ku 2V Equivalently, V is a subspace if au+bv 2V for all a;b2R and u;v 2V. (You should try to prove that this is an equivalent statement to the rst.)
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