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final_review - MA 35100 LECTURE NOTES FINAL EXAM REVIEW 4.1...

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MA 35100 LECTURE NOTES: FINAL EXAM REVIEW § 4.1: Introduction to Linear Spaces (cont’d) Summary 4.1.6 To find a basis for a linear space V , perform the following: #1. Choose an arbitrary element f V . #2. Find a list of elements f 1 , f 2 , . . . , f n V and constants c 1 , c 2 , . . . , c n such that f = c 1 f 1 + c 2 f 2 + · · · + c n f n . Then B = ( f 1 , f 2 , . . . , f n ) spans V . #3. Verify that the elements f 1 , f 2 , . . . , f n are linearly independent i.e., find all solutions to the equation c 1 f 1 + c 2 f 2 + · · · + c n f n = 0 . Then B = ( f 1 , f 2 , . . . , f n ) is a basis for V . Definition 4.1.8 A linear space V is called finite dimensional if it has a finite basis B = ( f 1 , f 2 , . . . , f n ) , so that we can define dim ( V ) = n . Otherwise, the space is called infinite dimen- sional. § 4.2: Linear Transformations and Isomorphisms Definition 4.2.1 Consider two linear spaces V and W . a. A function T : V W is called a linear transformation if it preserves linear combi- nations. Explicitly, for all f, g V and scalars k , i. T ( f + g ) = T ( f ) + T ( g ) . ii. T ( k f ) = k T ( g ) . b. The kernel of T is ker ( T ) = { f V : T ( f ) = 0 } . It is a subset of the domain V . c. The image of T is im ( T ) = { g W : g = T ( f ) for some f V } . It is a subset of the codomain W . Theorem. Let T : V W be a linear transformation. Then ker ( T ) is a subspace of V , and im ( T ) is a subspace of W . Definition. Let T : V W be a linear transformation. The rank of T is dim ( im ( T )) , and the nullity of T is dim ( ker ( T )) . Theorem. Let T : V W be a linear transformation between two finite dimensional vector spaces. Then rank ( T ) + null ( T ) = dim ( V ) . Definition 4.2.2 Let T : V W be a linear transformation. We say that T is an isomorphism if T is invertible. Theorem 4.2.3 Let V be a finite dimensional linear space with basis B = ( f 1 , f 2 , . . . , f n ) . The B -coordinate transformation L B : V R n , f = c 1 f 1 + c 2 f 2 + · · · + c n f n 7→ [ f ] B = c 1 c 2 . . . c n . is an isomorphism. Theorem 4.2.4 Let T : V W be a linear transformation between two finite dimensional vector spaces. Then the following are equivalent. i. T is an isomorphism. ii. ker ( T ) = { 0 } and im ( T ) = W . iii. null ( T ) = 0 and dim ( V ) = dim ( W ) . 1
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§ 4.3: Matrix of a Linear Transformation Definition 4.3.1 Let V be an n -dimensional linear space with basis B . Given a linear trans- formation T : V V , the unique matrix B such that [ T ( f )] B = B [ f ] B for all f V is called the B -matrix of T .
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