three_iv.pdf - Three.IV Matrix Operations Linear Algebra...

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Three.IV Matrix Operations Linear Algebra Jim Hefferon
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Sums and Scalar Products
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Representing operations on linear functions Recall that the collection of linear functions between two spaces L ( V, W ) is a vector space. That is, given a linear map f : V W then the scalar multiple rf ~ v r · f 7-→ r · ( f ( ~ v ) ) is a linear map rf : V W . And, where f, g : V W are linear then their sum ~ v f + g 7-→ f ( ~ v ) + g ( ~ v ) is also linear f + g : V W .
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Representing operations on linear functions Recall that the collection of linear functions between two spaces L ( V, W ) is a vector space. That is, given a linear map f : V W then the scalar multiple rf ~ v r · f 7-→ r · ( f ( ~ v ) ) is a linear map rf : V W . And, where f, g : V W are linear then their sum ~ v f + g 7-→ f ( ~ v ) + g ( ~ v ) is also linear f + g : V W . We now see how the matrix representation of Rep B,D ( f ) is related to that of Rep B,D ( rf ) , and how the representations of Rep B,D ( f ) and Rep B,D ( g ) combine to give the representation of Rep B,D ( f + g ) .
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Example Fix a domain V and codomain W with bases B = h ~ β 1 , ~ β 2 i and D = h ~ δ 1 , ~ δ 2 i . Let f : V W be the linear map represented by a matrix H We will find the matrix representing the map 6f : V W . ~ v 6f 7-→ 6 · f ( ~ v ) Let this be the representation of an output vector f ( ~ v ) . Rep D ( f ( ~ v ) ) = w 1 w 2 Note that 6 · f ( ~ v ) = 6 · ( w 1 ~ δ 1 + w 2 ~ δ 2 ) = ( 6w 1 ) · ~ δ 1 + ( 6w 2 ) · ~ δ 2 , so 6f ( ~ v ) has this representation. Rep D ( 6f ( ~ v )) = 6w 1 6w 2 Entries of the representation of 6f ( ~ v ) are 6 times as big as the entries of the representation of f ( ~ v ) . So the matrix representing 6f needs entries that are 6 times as big as those of the matrix representing f .
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For example, if f is represented by this matrix H = Rep B,D ( f ) = 2 1 3 4 then this is the matrix-vector representation of the action of f . 2 1 3 4 v 1 v 2 = 2v 1 + v 2 3v 1 + 4v 2 and this is the matrix-vector representation of the action of 6f . Rep B,D ( 6f ) · v 1 v 2 = 12v 1 + 6v 2 18v 1 + 24v 2 So this is the matrix representing 6f . Rep B,D ( 6f ) = 12 6 18 24
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Example Next consider the representation of the sum f + g of two linear maps f, g : V W . For a domain vector ~ v let the outputs f ( ~ v ) and g ( ~ v ) have these representations. Rep D ( f ( ~ v )) = u 1 u 2 Rep D ( g ( ~ v )) = w 1 w 2 The action of f + g is this. ~ v f + g 7-→ ~ u + ~ v = ( u 1 ~ δ 1 + u 2 ~ δ 2 ) + ( w 1 ~ δ 1 + w 2 ~ δ 2 ) = ( u 1 + w 1 ) · ~ δ 1 + ( u 2 + w 2 ) · ~ δ 2 The effect on the representations of adding the functions is to add the column vectors. Rep D ( ( f + g ) ( ~ v ) ) = u 1 + w 1 u 2 + w 2 Therefore, the matrix representing f + g needs entries that are the sum of those of the two matrix representations.
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For instance, suppose that the maps are represented by these. Rep B,D ( f ) = 2 1 3 4 Rep B,D ( g ) = 5 8 7 6 The functions given have this effect. Rep D ( f ( ~ v )) = 2 1 3 4 v 1 v 2 = 2v 1 + v 2 3v 1 + 4v 2 Rep D ( g ( ~ v )) = 5 8 7 6 v 1 v 2 = 5v 1 + 8v 2 7v 1 + 6v 2 The f + g matrix acts in this way Rep D ( ( f + g ) ( ~ v ) ) v 1 v 2 = 7v 1 + 9v 2 10v 1 + 10v 2 so this is its matrix representation.
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