MATH 2030: MATRICES. Av = 3


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1 MATH 030: MATRICES Introduction to Linear Transformations We have seen that we ma describe matrices as smbol with simple algebraic properties like matri multiplication, addition and scalar addition In the particular case of matrivector multiplication, ie, A = b where A is an m n matri and, b are n matrices (column vectors) we ma represent this as a transformation on the space of column vectors, that is a function F () = b, where is the independent variable and b the dependent variable In this section we will give a more rigorous description of this idea and provide eamples of such matri transformations, which will lead to the idea of a linear transformation To begin we look at a matrivector multiplication to give an idea of what sort of functions we are working with A = 0, v = 3 4 then matrivector multiplication ields Av = 3 [ ] We have taken a matri and produced a 3 matri More generall for an [ we ma describe this transformation as a matri equation 0 = From this product we have found a formula describing how A transforms an arbitrar vector in R into a new vector in R 3 Epressing this as a transformation T A we have T A = From this eample we can define some helpful terminolog A transformation T from R n to R m is a rule that assigns to each each vector v R n a unique vector T (v) R m The domain of T is R n and the codomain is R m, and we write this as T : R n R m For a vector v in the domain of T, the vector in the codomain T (v) is called the image of v under T The set of all possible images T (v) for all vinr n is called the range of T In the previous eample the domain of T A is R and the or mapping or function
2 MATH 030: MATRICES [ codomain is R 3, so T A : R R 3 The image of v = is w = T (v) = 3 ] The image of T A consists of all vectors in the codmain of the form T A = this describes an arbitrar linear combination of the column vectors of A We conclude that the image consists of the column space of A Geometricall we ma see this as a plane in R 3 through the origin with the column vectors of A as direction vectors Notice that T A () R 3 where is an vector in R Linear Transformations The previous eample T A is a special case of a more general tpe of transformation called a linear transformation We provide a less rigorous definition, that summarizes the ke ideas that the transformation respect vector operations of addition and scalar multiplication Definition 0 A transformation T : R m R m is called a linear transformation if () T (u + v) = T (u) + T (v) for all u and v in R n () T (cv) = ct (v) for all v in R n and all scalars c Eample 0 Consider once again the transformation T : R R 3 defined b T = [ w we will show this is indeed a linear transformation Define u = and v = z then compute T (u + v), T ( + ) w = T z () + w = + z + w + w ( + w) 3( + z) = ( 3) + (w 3z) 3( + w) + 4( + z) (3 + 4) + (3w + 4z) Looking at the farright hand side we ma write this as w [ ( 3) + (w 3z) w = T + T = T (u) + T (v) z (3 + 4) (3w + 4z) To show the second propert, consider T (cv) for some scalar c: ( [ () c c c T c = T = c c = c( ) = c ) c 3c + 4c c(3 + 4) the second propert holds, this is indeed a linear transformation = c [ Although the linear transformation T in the previous eample arose as a matri transformation T A, one ma go backwards and recover the matri A from the
3 MATH 030: MATRICES 3 definition of T given in the eample Notice that [ 0 0 [ T = = + = where this is just the matrivector multiplication of A with an arbitrar vector in the domain In general a matri transformation is equivalent to a linear transformation, according to the net theorem Theorem 03 Let A be an m n matri Then the matri transformation T A : R n R m defined b is a linear transformation T A () = A, R n Proof Let u and v be vectors in the domain, and c a scalar, then T A (u + v) = Au + Av = T A (u) + T A (v) and T A (cv) = cav = ct A (v) Thus T A is a linear transformation Eample 04 Q: Let F : R R be the transformation that sends each point to its reflection in the ais Show that F is a linear transformation A: This transformations [ send ] each point (, ) to a new coordinate (, ), and so we ma write F = To show this is linear notice that [ = [ [ [ = ] 0 ] Thus F = A showing that this is a matri transformation and hence a linear transformation b the previous theorem Eample 05 Q:Let R : R R be the transformation that rotates each point b an angle of π/4 (90 degrees) counterclockwise about the origin Show that F is a linear transformation A: Plotting this on the plane, we see that R takes an point (, ) in the plane and sends it too (, ), and so as a transformation [ [ [ 0 0 R = = + = ] 0 0 So R is described b a matri transformation and therefore is a linear transformation Recalling that if we multipl a matri b standard basis vectors we find the columns of the original matri, we can use this fact to show that ever linear transformation from R n to R m arises as a matri transformation Theorem 06 Let T: R n R m be a linear transformation Then T is a matri transformation, and more specificall T = T A where A is the m n matri A = [T (bfe ) T (e ) T (e n )] Proof Let e, e,, e n be the standard basis vectors in R n and let be a vector in R n, so that = e + + n e n Noting that T (e i ) for i =,, n are column
4 4 MATH 030: MATRICES vectors in R m, we denote A = [T (bfe ) T (e ) T (e n )] be the m n matri with these vectors as its columns, then T () = T ( e + + n e n ) = [T (bfe ) T (e ) T (e n )] n = A The matri in the proof of the last theorem is called the standard matri of the linear transformation T Eample 07 Q: Show that a rotation about the origin through an angle θ defines a linear transformation from R to R and find its standard matri A: Let R θ be the rotation, we will prove this geometricall Let u and v be vectors in the plane, then the parallelogram rule determines the new vector u + v If we now appl R θ the parallelogram is rotated b an angle of θ and so the diagonal of the parallelogram defined b R θ (u) + R θ (v) Hence R θ (u + v) = R θ (u) + R θ (v) Similarl if we appl a rotation to v and cv b a fied angle of θ we find R θ (v) and R θ (cv), however as rotations do not affect lengths we must have R θ (cv) = cr θ (v) We conclude that R θ is a linear transformation, and we ma appl the standard basis vectors of R to this transformation to determine its standard matri Using trigonometr we find that [ R = cosθ sinθ Equivalentl we find that the second standard basis vector is mapped to [ 0 sinθ R = ] cosθ Thus the standard matri for R θ will be cosθ sinθ sinθ cosθ Eample 08 Show that the transformation P:R R that projects a point onto the ais is a linear transformation and find its standard matri More generall, if l is a line through the origin in R, show that the transformation P l : R R that projects a point onto l is a linear transformation and find its standard matri A: P sends the point (, ) to the point (, 0) and so [ [ [ [ [ 0 0 P = = + = 0 0 Thus the transformation matri for P is just 0 0 The line l has direction vector d, then for an vector v, the transformation P l is given b proj d (v)  the projection of v onto d, ( ) d v proj d (v) = d v v
5 MATH 030: MATRICES 5 To show P l is linear consider the sum ( ) d (u + bv) P l (u + v) = d v v ( ) d u + d v = d v v ( ) ( ) d u d v = d + d v v v v the last line is just P l (u) + P l (v) Similarl P l (cv) = cp l (v), proving that P l is indeed a linear transformation d To determine its standard matri, we denote d =, the projection d onto the standard basis is just P l (e ) = d d d + d d P l (e ) = d d d + d d impling that the standard basis is of the form d A = d d d + d d d d New Linear Transformations from Old If T:R m R n and S:R n R p are linear transformations, then we ma follow T b S to form the composition of the two transformations, denoted S T Notice that in order for S T to make sense, the codomain of T and the domain of S must be the same, and the resulting transformation S T goes from R m to R p, that is it maps from the domain of T to the codomain of S The formal definition of this new function is given as S T (v) = S(T (v)) We would like to have this new function be a linear transformation, which it is, and we ma demonstrate this b showing that S T satisfies the definition of a linear transformation We will do this b showing that it is a matri transformation Theorem 09 Let T: R m R n and S R n R p be linear transformations Then S T : R m R p is a linear transformation Moreover, their standard matrices are related b [S T ] = [S][T ] Proof Let [S] = A and [T ] = B, so that A is an m n matri and B a n p matri; if v is a vector in R m we simpl compute S T (v) = S(T (v)) = S(Bv) = A(Bv) = (AB)v Thus the effect of S T is to multipl vectors b AB, from which it follows immediatel that S T is a matri transformation and hence a linear transformation with the transformation rule [S T ] = [S][T ] Eample 00 Q:Consider the linear transformation T:R R 3 defined b T = 3 + 4
6 6 MATH 030: MATRICES and the linear transformation defined S R 3 R 3 defined b S + 3 = Find S T : R R 3 A: Calculating the matrices of each transformation and computing their product we find 0 [S T ] = [S][T ] = = It follows that the corresponding transformation is then (S T ) = [S T ] = Eample 0 Q: Find the standard matri of the transformation that first rotates a point 90 degrees counterclockwise about the origin and then reflects the result in the ais A: The rotation matri [R] and reflection matri [F ] were given in previous eamples as [R] = 0, [F ] = composing the two we find the desired transformation 0 [R R] = [F ][R] = 0 Inverse of Linear Transformations Consider the effect of a 90 degree counterclockwise rotation about the origin followed b a 90 degree clockwise rotation about the origin The cumulative effect of these two transformations is the identit transformation I, that is, no change at all (I(v) = v) If we denote R 90 and R 90 for the respective transformations this means (R 90 R 90 (v) = v for an v in R Reversing the order geometricall gives the same result as well, ie R 90 R 90 (v) = v as well Thus these two linear transformations are inverses of each other and we sa that an two transformations related in this manner are called inverse transformations Definition 0 Let S and T be linear transformations from R n to R n Then S and T are inverse transformations if S T = I n and T S = I n In terms of matrices, if S and T are inverse transformations then [S] = [T ] since [S][T ] = [S T ] = I where the last matri is the identit matrithis show that [T ] and [S] are inverse matrices Furthermore, if a linear transformation T is invertible, then its standard matri [T ] must be invertible as well As matri inverses are unique, this means that the inverse of T is also unique, therefore we can use the notation T to denote the unique inverse of each invertible linear transformation
7 MATH 030: MATRICES 7 Theorem 03 Let T:R n R n be an invertible linear transformation Then its standard matri [T ] is an invertible matri and [T ] = [T ] Eample 04 Q: Find the standard matri of a 60 degree clockwise rotation about the origin in R A: Putting θ = π/3 in the sines and cosines in the matri [R θ ] and using basic trig we find that [R 60 ] = 3 3 Using the fact that a 60 degree clockwise rotation is the inverse of R 60, and so we ma find that b appling the last theorem R 60 ] = [R 60 ] = 3 3 Eample 05 Q:Determine whether projection onto the ais is an invertible transformation, and if it is, find the inverse A: [ We] have seen that the standard matri for this projection transformation P is 0, this is not an invertible matri as its determinant vanishes We conclude 0 0 that P is not invertible as well References [] D Poole, Linear Algebra: A modern introduction  3rd Edition, Brooks/Cole (0)
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