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Linear Transformations

Linear Transformations. Fall 2009. Definitions. Let V and W be vector spaces. A function L:V → W is called a linear transformation of V into W if a) L( u + v ) = L( u ) + L( v ) b) L( c u ) = c L( u ) for u ε V and real c If V = W, then L is called a linear operator.

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Linear Transformations

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  1. Linear Transformations Fall 2009

  2. Definitions • Let V and W be vector spaces. A function L:V → W is called a linear transformation of V into W if a) L(u + v) = L(u) + L(v) b) L(cu) =cL(u) for uε V and real c • If V = W, then L is called a linear operator

  3. Example • Define a mapping L: R3→ R2 as • To verify, letbe arbitrary

  4. Example • Let K( x,y ) be continuous in x and y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. Define L: C [0,1] → C [0,1] as From the properties of integrals, conditions (a) and (b) hold

  5. Example • Define a mapping L: R3→ R3 as L is a linear operator on R3. If r > 1, it is called a dilation. If 0 < r < 1, it is called a contraction. General term is scaling

  6. Example • Define a mapping L: R3→ R3 as L is a linear transformation. More generally, if A is an mxn matrix, then L(x)=Ax is a linear transformation from Rn to Rm

  7. Definition and Examples Example • Define a mapping L: R3→ R3 as L is a linear transformation. More generally, if A is an mxn matrix, then L(x)=Ax is a linear transformation from Rn to Rm

  8. Definition and Examples Example • Define a mapping L: R3 R3 as L is a linear transformation. More generally, if A is an mxn matrix, then L(x)=Ax is a linear transformation from Rn to Rm

  9. Example • Define a mapping L: R3→ R3 as Let

  10. Example • Define a mapping L: R2 R2 as Let u= [ u1u2 ], v= [ v1v2 ] be in R2 So L is not a linear transformation

  11. Definitions • A linear transformation L: V  W is one to one if it is a one to one function, i.e. if v1 ≠ v2 implies L(v1) ≠ L(v2). (Equivalently, L is one to one if L(v1) = L(v2) implies v1=v2.) • Let L: V  W be a linear transformation. The kernel of L, ker L, is the subset of V consisting of all vεV such that L(v) = 0W • Comment - Since L(0V) =0W, ker L is not empty

  12. Kernel and Range of a Linear Transformation • Define L: P2 R as i) Find ker L ii) Find dim ker L iii) Determine if L is one to one

  13. Example (continued) i) So ker L consists of polynomials of the form

  14. Example (continued) ii) So the vectors ( t2 - 1/3 ) and ( t - 1/2 ) span ker L. You can argue that they are linearly independent. So the set { t2 - 1/3, t - 1/2 } is a basis for ker L. iii) Since dim ker L = 2, L is not one to one • By theorem: L is one to one if and only if dim ker L = 0

  15. Kernel and Range of a Linear Transformation • Let L: R3 R3be defined by a) Is L onto? b) Find basis for range L c) Find ker L d) Is L one to one?

  16. Example (continued) a) Let be arbitrary. Find such that L(v) =w Solution exists only if c–b–a= 0 So, L is not onto

  17. Example (continued) b)Range of L is the span of You can show that the first two vectors are linearly independent and the third is the sum of the first two. Alternatively, you could take the transpose of the matrix and put it into row echelon form to get a basis for the row space of the transpose. Either way, the basis is

  18. Example (continued) c) Kernel of L consists of all vectors such that L(v) =0 Set a3=r, then a1=– r and a2=– r. So, all vectors in the kernel look like Basis for ker L is

  19. Example (continued) d) To see if L is one to one, let with v ≠ w. Is it possible to have L(v) = L(w)? L(v) = L(w)  L(v) - L(w) =0 L(v - w) =0 So, v - wker L (null space of the matrix) and • Note that dim ker L + dim range L = dim domain L Since it is possible to have L(v) = L(w) when v ≠ w, L is not one to one

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