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.(3)Ifu V, thenu= (t,3t, 2t)for somet R, so ifk R, thenku= (kt,3(kt), 2(kt)) = (t ,3t , 2t ) Vwheret =kt R. Example unit circleS1inR2is not a subspace because it doesn t contain0= (0,0)andbecause, for example,(1,0)and(0,1)lie inSbut(1,0) + (0,1) = (1,1)does not. Similarly,(1,0)lies inSbut2(1,0) = (2,0)does not. Alinear combinationof vectorsv1.
span(S) := fv 2Rnjv is a linear combination of vectors in Sg (0.2) The span of the empty set, ?, is by de nition ... The dimension of a subspace V of Rn is the cardinality of any basis for V, ... But this contradicts linear independence, since 1 6= 0. Hence v k cannot be a linear combination of the other v k. By re-indexing the v
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