In this chapter you will learn how to use MATLAB to work with lengths, angles and projections in subspaces of R and later in certain linear spaces.

Transcription

1 Chapter 5. Introduction In this chapter you will learn how to use MATLAB to work with lengths angles and n projections in subspaces of R and later in certain linear spaces.. Lengths and Angles Recall from class that the dot product is your key for understanding lengths and angles. The best way to find the dot product of following: u = and v = 9 in MATLAB is to do the >> u = [; ; ]; >> v = [; ; 9]; >> dot(uv) Recall that the length of u is denoted by u calculated by computing the angle between u and v then u v = u v cosθ. u u and if θ is Here are some other MATLAB commands that will help you get through this section. >> sqrt(9) %finds the square root of 9 >> acos(/5) %finds the inverse cosine of /5 (Do not use the MATLAB function norm) a. 5. # b. 5. #5. Orthogonal Projections In this section you will use MATLAB to project a vector onto a subspace. For now the subspace will be defined by an orthonormal basis. In section 7 you will see another way to do this even if you do not have an orthonormal basis.

2 n Here is the key idea: Assume V is a subspace of R u u u... u ) is an orthonormal ( n basis for V and x is any vector in R n. Since the orthogonal projection of x onto V is in V and we have a basis for V we only need to find the correct coefficients to use to build the projection. More specifically we need to find the correct c c... cn such that projv ( x) = c u + cu cnun. Here is the secret: c = x u c = x u c = x n u n Here is how you would do 5. #8 Find the orthogonal projection of onto the subspace of R spanned by >> v = [; ; ; ]; >> v = [; ; -; -]; >> v = [; -; -; ]; >> u = (/norm(v))*v; %Normalize the vectors >> u = (/norm(v))*v; >> u = (/norm(v))*v; >> x = [;;;]; >> c = dot(xu); %Calculate the coefficients >> c = dot(xu); >> c = dot(xu); >> proj = c*u + c*u + c*u; %Calculate the projection >> rats(proj) %Display projection with rational numbers / / -/ /

3 So the orthogonal projection of. onto the subspace spanned by is a. 5. # b. 5. #9 (Hint: Look at Fact 5..9). Roll Your Own Orthonormal Basis n The Gram-Schmidt process is a way to turn any basis in R into an orthonormal basis. We are going to start with a basis ( v v... vn ) and turn it into the orthonormal basis u u... u ) that spans the same space. Here is the basic idea: ( n ) If v is not a unit vector divide by its magnitude to turn it into a vector with length. Call this new vector u. ) v has a component parallel to u and a component perpendicular to u. To get u just normalize the perpendicular component. ) v has a component in the subpace spanned by ( u u ) and a component orthogonal to the subspace. u is the normalized perpendicular component. If you keep doing step ) on all of your vectors you will end up with an orthonormal basis that spans the same space. Here is how you would do this in MATLAB starting with the basis. MATLAB trick: The subplot command lets you put several graphs in the same window. The following is somewhat long but worth typing in.

4 >> v = [;-]; >> v = [;]; >> origin = [;]; >> picture_ = [v origin v]; >> subplot(); >> plot(picture_(:)picture_(:)'linewidth'); >> axis([- - ]); >> axis('square'); >> u = (/norm(v))*v; >> picture_ = [u origin v]; >> subplot(); >> plot(picture_(:)picture_(:)'linewidth'); >> axis([- - ]); >> axis('square'); >> v_parallel = dot(vu)*u; >> v_perp = v - v_parallel; >> picture_ = [u origin v_perp]; >> subplot(); >> plot(picture_(:)picture_(:)'linewidth'); >> axis([- - ]); >> axis('square'); >> u = (/norm(v_perp))*v_perp; >> picture_ = [u origin u]; >> subplot(); >> plot(picture_(:)picture_(:)'linewidth'); >> axis([- - ]); >> axis('square');>> u %see the value of u u = >> u %see the value of u u = The new basis is a. 5. # (you do not need to plot your vectors for this one)

5 b. 5. #9 c. 5. # 5. Transposes and Symmetry Oftentimes given a matrix it is useful to make a new matrix whose columns are the rows of the original matrix. This new matrix is called the transpose of the original matrix. More formally the entry in the i th row of the j th column of A is the j th row and i th column of the transpose of A. The ' command makes this easy to do in MATLAB. >> A = [ ; 9 7 5] A = >> A' Recall T ) A is a symmetric matrix iff A = A ) A is a skew-symmetric matrix iff A = A T Use symbolic x matrices in MATLAB to answer the following problems. Then generalize (without MATLAB) to the n x n case. Hints: ( A T A) T T T T ( AB ) = B A. A T ) T = A T T = A ( A ) T = A ( Here is how to prove that A T A T A is symmetric: 5a. 5. # 5b. 5. # 5

6 . Orthogonal Matrices There are several things you know about orthogonal transormations and you can use all of them to help decide whether a given matrix is orthogonal. What you know: ) An orthogonal transformation preserves vector lengths (that is T is orthogonal iff T ( x) = x for all x ) An orthogonal transformation preserves angles between vectors. ) The columns of the matrix form an orthonormal basis which means that A T A = I. (Do you see why? Lok at Fact 5..7) Here is how to use MATLAB to check each of these properties. Note that ) and ) can only be used to show that a transformation is not orthogonal while ) can be used to show that a matrix is orthogonal and that can be used to show that a matrix is not orthogonal. ) Use property : >> A = [ ; ]; >> x = [;]; >> norm(x). >> norm(a*x). So x A x and hence is not orthogonal. ) Use property : (This is 5. #) >> A = (/7)*[ -; - ; ]; >> v = [;;]; >> v = [;;]; >> acos(dot(vv)/(norm(v)*norm(v)))

7 .775 >> acos(dot(a*va*v)/(norm(a*v)*norm(a*v))).5 So then angle between v and v is not the same as the angle between Av and Av. Hence 7 is not orthogonal since ) Use property : (This is 5. #) A T A = I. >> A = (/)*[ - ; ; -]; >> A'*A So is orthogonal. a. 5. # b. 5. # c. 5. #5 7. Least Squares Solutions and Another Way to Calculate Orthogonal Projections. In a perfect world every set of equations would be consistent. Unfortunately our world is not perfect. Luckily a theory has been developed to come up with the best solution to an inconsistent set of equations. More specifically if Ax = b is inconsistent we solve for Ax = the projection of b onto im(a). A way of finding x is given below. Once you know x computing Ax will give you the projection. 7

8 We will do 5. example in MATLAB. The set of equations we want to solve is x + y = x + y = x + y = Note that this system is inconsistent. We will also use the best solution to orthogonally project >> A = [ ; ; ]; >> b = [; ; ]; >> x = inv(a'*a)*a'*b x = -.. >> A*x onto im(a) So the least squares solution is image of is 5 x * = and the orthogonal projection of onto the 7a. 5. # (You can use MATLAB instead of a paper and pencil) 7b. 5. # 8

9 8. Data Fitting In this section you will use section 7 to find the best equation to fit some given data. Here is 5. #: We are trying to fit a linear function of the form f ( t) = c + ct to the data points ( ) ( ) and ( ). In other words we want to find a solution to the following equations: = c + (since we want the point ( ) on the line) c = c + (since we want the point ( ) on the line) c = c + (since we want the point ( ) on the line) c In matrix form = c c >> A = [ ; ; ]; >> b = [;;]; >> x = inv(a'*a)*a'*b x =..5 So our linear function is f ( t) = +. 5t. We will use MATLAB to see how good it is. Remember that our three points are ( ) ( ) and ( ). MATLAB trick: The scatter command will plot points without connecting them. >> points = [ ; ] points = >> scatter(points(:) points(:) 'filled'); >> axis([- 7]) %Set the axis >> hold on %Stop MATLAB from erasing our graph 9

10 >> t = linspace(-); %Set up domain for linear function >> y = +.5*t; %Define linear function >> plot(ty); %Be sure to look at the graph More problems: 8a. 5. # 8b. 5. # Hint: Here is how to plot the quadratic function y = t + t + 5 on the interval [ ] >> t = linspace(-); >> y = -*t.^ + *t + 5; Note the. in front of the ^ >> plot(ty); 8c. 5. #8 9. Inner Product Spaces This is a good place to look back and see what you ve accomplished in the last couple of chapters. Here s a recap: n Chapter : Useful concepts forr including linear independence dimension and subspace among others. Chapter : The Chapter concepts generalized to arbitrary linear spaces. n 5. 5.: dot products lengths angles and orthogonal projections for R. Now we are going to look at dot products lengths angles and orthogonal projections for some spaces other than R n. The basic idea is that given a linear space we make up a rule that has the same properties as the dot product on R n. This rule is called an inner product. Here is 5.5 example 5: The space is C [ π ] and we define < f g >= f ( t) g( t) dt. Some notes about MATLAB before we do this: π ) trapz is the MATLAB command to use the trapezoid rule of integrating. ) MATLAB is subject to round off errors. If MATLAB reports that something is close to it is probably actually equal to for the applications we are doing.

11 >> t = linspace(*pi); >> trapz(tsin(t).*cos(t)).e-7 π 7 The last line says that f ( t) g( t) dt =. =. which is close enough to for us to call it equal to. 9a. 5.5 # 9b. 5.5 # 9c. 5.5 #. Orthogonal Projections Once you have an inner product you can do orthogonal projections like you did in section above. We will go through 5.5 example 8. We are using the formula from Fact 5.5.5: f n ( t) = a + b sin( t) + c cos( t) bn sin( nt) + cn cos( nt) where a = π π π f dt b n c n π π = π f ( t) sin( nt) dt π π = π f ( t) cos( nt) dt >> t = linspace(-pipi); %set up the domain >> f = t; %define the function >> %calculate the coefficients >> a = (/pi)*trapz(t /sqrt()*f); >> b = (/pi)*trapz(t sin(t).*f); >> c = (/pi)*trapz(t cos(t).*f); >> b = (/pi)*trapz(t sin(*t).*f); >> c = (/pi)*trapz(t cos(*t).*f); >> b = (/pi)*trapz(t sin(*t).*f); >> c = (/pi)*trapz(t cos(*t).*f); >> b = (/pi)*trapz(t sin(*t).*f); >> c = (/pi)*trapz(t cos(*t).*f);

12 >> %compute the approximations >> f = a*(/sqrt()) + b*sin(t) + c*cos(t) >> f = a*(/sqrt()) + b*sin(t) + c*cos(t) + b*sin(*t) + c*cos(*t); >> f = a*(/sqrt()) + b*sin(t) + c*cos(t) + b*sin(*t) + c*cos(*t) + b*sin(*t) + c*cos(*t); >> f = a*(/sqrt()) + b*sin(t) + c*cos(t) + b*sin(*t) + c*cos(*t) + b*sin(*t) + c*cos(*t) + b*sin(*t) + c*cos(*t); >>%graph the approximations with the original function >> subplot(); plot(tf); hold on; plot(tf); >> subplot(); plot(tf); hold on; plot(tf); >> subplot(); plot(tf); hold on; plot(tf); >> subplot(); plot(tf); hold on; plot(tf); a. 5.5 # b. 5.5 #