Alg1231W6T1 - Introduction to linear maps

Sunday, 2 September 2018

3:43 PM

Machine generated alternative text: Not every function is linear. However, the special class of linear functions turn out to be so useful in practice that it is worth spending an entire chapter studying them (or even an entire course) Considerftvo vector space V and W, both over the scalar field F. A function, or map; or transformation T : V —+ W beftveen these vectors spaces is linear if it satisfies both the Addition Condition: T(VI + v2) = T(VI) + T(V2) for all VI, v2 e V, Cl In(x + y) = In(x) + In(y) and the Scalar Multiplication Condition: T(Av) Below is a list of equations of the form AT(v) for all v e V and all scalars e F. ranging over quite a few different contexts. Which of these equations are valid? In these cases S is said to satisfy the addition condition. Cl Ix+yl = + 1 1 g@) dc Cl sin@ + y) = sin@) + sin(y) 2 df(x) dc dg(x) dc Note: a • x above refers to the dot product and a x x refers to the cross product. Below is a list of equations of the form S(Äu) = AS(u). Which of these equations are valid? In these cases S is said to satisfy the scalar multiplication conditiom sin(Äx) = Asin(x) IAxl = .\lxl Af(x) dc dc a • (Ax) = .X(a •x

 

Machine generated alternative text: A linear map T between vectors space V and W behaves quite nicely. Two well known properties shared by all linear maps are: The zero vector in V maps to the zero vector in W, i.e T(O) = O 2 The negative of a vector maps to the negative of the corresponding function value; i.e. T(—v) These properties are also shared by some non-linear functions in calculus, such as odd functions. Which of the following satisfy T(O) = 09 cos@) Which of the following satisfy cos@) -T(vp

 

Machine generated alternative text: Which of the following satisfy both properties above, but are NOT an examples of linear maps? cos@)

 

Machine generated alternative text: One of the lovely properties of a linear map T on a vector space like IRn (distinguishing it from other arbitrary functions) is that we only need to know what T does to a small number of linearly independent vectors to define the entire map. For example; consider a linear map T from R to R 2 Then it would be easy to work out the value of T( 5 where we know 1 3 —9 -2 and T( 1 ) since this must be equal to the linear combination 1 2 3 3 4 7 1 Using this idea we calculate that Note: the Maple notation for the vector 1 2 3 is <

 

Machine generated alternative text: Let {VI , v2} be linearly independent vectors in a real vector space V and let v3 T(VI) ¯ i) If we assume T is a linear map, then we can calculate T(V3) ¯ 1 Note: the Maple notation for the vector is < 2 2v1 + v2 . Consider a map T where we know 1 and T(V2) ¯ 5 -2 2 ii) Subsequently we are told that T(V3) Q Is this consistent with our assumption that Twas a linear map? No. 11

 

Machine generated alternative text: Let F(t) be a function oft > O . The Laplace Transform of F is a new function in the variable s: The Laplace transform is a linear map between the vector space of functions in t to the vector space of functions in s. Let's prove that it satisfies the properties required of a linear transformationThere are ftvo conditions to be satisfied. Consider F(t) and G(t) _ We have Now because the integral of a sum is the sum of the integrals the right hand side can be rewritten as o the right hand side can be rearranged as COE) = A stdt, and by definition of the Laplace Transform this is equivalent to O C(F+G) @ C(F+G) O C(F+G) O C(F+G) Hence the Laplace transform satisfies the Now consider F(t) and e R . We have Since constants factor out of integrals addition condition O

 

Machine generated alternative text: By the definition of the Laplace Transform this is equivalent to o C(ÄF) = C(F) + C(G) o C(ÄF) = + C(F) @ C(ÄF) - o C(ÄF) = C(F). Hence the Laplace Transform also satisfies the scalar multiplication condition linear transformations also play an important role in calculus. Pierre-Simon Laplace 1749-1872 O , and we can conclude that the Laplace Transform is a linear map. So

 

 

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