1、Linear Transformations3rd week / Linear AlgebraObjectives of This Week2The goal is to understandBasics of transformation Linear transformationStandard matrixOnto and one-to-one3 A transformation, function, or mapping, maps an input to an output Mathematical notation: : Domain: Set of all the possibl
2、e values of Co-domain: Set of all the possible values of Image: a mapped output , given Range: Set of all the output values mappedby each in the domain Note: the output mapped by a particular is uniquely determined. Transformation123412345678910A BRange +Domain Codomain Definition: A transformation
3、(or mapping) is linear if: Simple example: : , = = 34Linear TransformationI. + = + for all , in the domain of and for all scalars and : 31 = 1 1 = 3 : 32 = 2 2 = 6: 341 +52 41 +52= 14 = 42: 31 = 1 1 = 3: 32 = 2 2 = 6 4 1 +5 2= 12+30 = 425 :x y : Mapping -dim vector to -dim vector Example: Transforma
4、tions between Vectorsx =123 3 y = x = 45 2:x 3 y 26 Example: Suppose is a linear transformation from 2 to 3 such that 10 =211and 01 =012. With no additional information,find a formula for the image of an arbitrary x in 2Matrix of Linear Transformationx = 12 = 1 10 +2 01 x = 1 10 +2 01 = 1 10 +2 01=
5、1211+2012=2 01 11 2127 In general, let be a linear transformation. Then is always written as a matrix-vector multiplication, i.e., x = x for all x In fact, the -th column of is equal to the vector e , where e is the -th column of the identity matrix in : = e1 e Here, the matrix is called the standar
6、d matrix of the linear transformation Matrix of Linear Transformation8 Example: Find the standard matrix of a linear transformation from 3 to 2 such that100= 12 ,010= 43 , and 001= 56 . Matrix of Linear Transformationx =123= 1100+2010+3001 x = 1100+2010+3001= 1100+2010+3001= 1 12 +2 43 +2 56 = 1 4 5
7、2 3 6123= xONTO and ONE-TO-ONE Definition: A mapping is said to be onto if each b is the image of at least one x . That is, the range is equal to the co-domain. is NOT onto is onto 9ONTO and ONE-TO-ONE Definition: A mapping is said to be one-to-one if each b is the image of at most one x . That is,
8、each output vector in the range is mapped by only one input vector, no more than that. is NOT one-to-one is one-to-one 0 0 0 01011 Fully-connected layersNeural Network ExampleWeightHeightIs_smokingover-weightedTall_and_smokingLife-spanx y1 z212 Will there be many (or unique) people mapped to the sam
9、e (over_weighted, tall_and_smoking)?Neural Network Example: ONE-TO-ONEWeightHeightIs_smokingover-weightedTall_and_smokingLife-spanx y1 z213 Is there any (over_weighted, tall_and_smoking) that does not exist at all?Neural Network Example: ONTOWeightHeightIs_smokingover-weightedTall_and_smokingLife-sp
10、anx y1 z214 Let be a linear transformation, i.e., x = x for all x . is one-to-one if and only if the columns of are linearly independent. maps onto if and only if the columns of span . ONTO and ONE-TO-ONE15 Example:Let x = x =2 01 11 212 Is one-to-one? Does map 2 onto 3?ONTO and ONE-TO-ONE16 Example:Let x = x = 1 4 52 3 6123 Is one-to-one? Does map 3 onto 2?ONTO and ONE-TO-ONELinear transformationStandard matrix Onto and one-to-oneSummary
