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Unformatted text preview: MA 35100 LECTURE NOTES: MONDAY, FEBRUARY 22 Linear Combinations and Span Last time, we made the following definition: Definition. The image of a linear transformation T : R m R n consists of all values ~ b = T ( ~x ) in the codomain: im( T ) = n ~ b R n : ~ b = T ( ~x ) for some ~x R m o R n . Moreover, if A is the matrix of T , we denote im( A ) = im( T ). Rather explicitly, since T is a linear transformation, there is an n m matrix A such that T ( ~x ) = A~x . Lets write the columns of A as ~v j i.e., A = ~v 1 ~v 2 ~v m . (Theorem 2.1.2 on page 47 of the text states ~v j = T ( ~e j ) R n .) The image of this linear transfor- mation consists of all vectors in the form ~ b = T ( ~x ) = A~x = x 1 ~v 1 + x 2 ~v 2 + + x m ~v m . We recall Definition 1.3.9 on page 30 of the text: Definition. A vector ~ b R n is called a linear combination of the vectors ~v 1 , ~v 2 ,..., ~v m R n is there exist scalars x 1 , x 2 ,..., x m such that ~ b = x 1 ~v 1 + x 2 ~v 2 + + x m ~v m . We use this to make a new definition: Definition. The set of all linear combinations of the vectors ~v 1 , ~v 2 ,..., ~v m R n is called their span : span( ~v 1 ,..., ~v m ) = n ~ b = c 1 ~v 1 + + c n ~v n : c 1 ,..., c m R o We have proven the following: Theorem. The image of a linear transformation T : R m R n is the span of the column vectors of its matrix A . Properties of the Image We state a property we will need for the next lecture: Theorem. Let T : R m R n be a linear transformation....
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