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Lecture 25

MATH125 Lecture 25: 25

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Department
Mathematics
Course
MATH125
Professor
Nikita
Semester
Winter

Description
Lecture 25, March 15, 2017. De▯nition. A transformation T : R ! R m is called linear if n (1) T(u + v) = T(u) + T(v) for all vectors u;v in R ; (2) T(cv) = cT(v) for all scalars c and vectors v in R . 2 3 Example. Consider the above transformation T : R ! R given by [ ] ▯ ▯ v v1 1 ▯! Av = ▯ 2v1▯ v2 ▯: v2 3v1+ 4v 2 2 Let us check that it is linear. Indeed, let u;v be two vectors in R . Then T(u + v) = A(u + v) = Au + Av = T(u) + T(v). Similarly, T(cv) = A(cv) = c(Av) = cT(v). In the above example there is nothing special in size 3 ▯ 2 of the matrix A. The same is true in general. Theorem. Let A be a matrix of size m ▯ n. Then the transformation TA: R ! R m given by v ! TA(v) = Av is a linear transformation. 2 2 Example. Let F : R ! R be the transformation that sends each point to its re ection in the x-axis. Let us show that it is a linear transformation. By construction, the transformation F sends the point (x;y) to the point (x;▯y). Thus we may write ([ ]) [ ] x x F y = ▯y : We could check that F is linear by verifying all conditions in the de▯- nition. B[t it ]s fa[ter]to o[serve]tha[ ][ ] x 1 0 1 0 x = x + y = : ▯y 0 ▯1 0 ▯1 y Thus ([ ]) [ ] x x F y = A y where [ ] 1 0 A = 0 ▯1 : It now follows from the above theorem that F is linear. Let us show that any linear transformation T : R ! Ris a matrix multiplication. Given such T we attach the matrix A = [T(e )T(e ):::T(e )] 1 2 n of size m ▯ n where 1 2e ;:::ne is the standard basis. Theorem. Let T be a linear transformation as above. Then T(v) = Av n for each vector v in R . Proof. Let ▯ ▯ v1 ▯ v2 ▯ v = ▯ . ▯ : . v n Clearly we have v = v1 1+ v2 2+ ▯▯▯ + vn n. Then with the use of linearity of T we have T(v) = T(v 1 1 v 2 2 ▯▯▯ + vn n) = v1T(e1) + ▯▯▯ +nv Tne ) = Av and we are done. ▯ De▯nition. The above matrix A is called the standard matrix of the linear transformation T. 2 2 Example. Let T : R ! R be the transformation that projects a point onto x-axis. Find the standard matrix of T. Solution. This transformation sends a point (x;y) to (x;0). Hence ([ 1 ]) [ 1 ] T(e1) = T = 0 0 and ([ ]) [ ] T(e 2 = T 0 = 0 : 1 0 Hence the required matrix is [
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