MAT224H1 Lecture Notes - Lecture 5: Brevican
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If w is a subspaceof rh defined bysolution set to homogeneous system of linear equations. X 2 t x3 t x4 x5 o of free variables. Let wi wz. be subspaces finite dimensional of u dimlwitdimlwd dimlw. nu then dimlw. tw z. Result obvious if either wi wz is to. 1 is a basis for w a basisfor wz is. If wz to wzewiawzwz. es wzesparhzhspanld sb pan as 4 swzzo swi. S independenceof s t get. Dim spans tdimlspantdtdimlspany dimlunwzlfd. im w _dimwinwz. lt ldimwz dimwnwzj dimwitdimwz dimw. nu. O v if a u btg. abc lr. nu eu. Ua anelr v vuev this is about n vectors not just 2 vectors so we cannot just assume it we need to prove it by induction. Practice i determine if thefollowingtransformations are linear define t da. Ctv c 0 t is called afia w n. Tfax axz. cn3 tcby. tbyztbyd a x y by yz. by. I fix dx j f x dx j i f 1.
