Practice Questions for test 2. ------------------------------ Ignore questions regarding linear transformations. 1.: Find a vector of length 5 perpendicular to (1,0,-1,2). 2.: Find the standard matrix form of the linear transformation L: R^3 -> R^2 which takes a vector to its projection onto the x-z-plane. 3.: Consider the 2 systems of parametric equations: (a) x=1+2t, y=2-3t, z=-1+t; (b) x=3+4s, y=-1-6s, z=2s. Do they describe the same line? Explain! 4.: Find the parametric equation of the line that passes through P=(2,1,-1) and is perpendicular to the plane E: x+2y+3z-4=0. 5.: We define a plane by E: (x,y,z) = (1,2,-1) + t(2,1,1) + s(1,0,2). (s and t are arbitrary real numbers) a) Find parametric equations of 2 lines that lie in the plane (you choose which lines!). b) Find a vector perpendicular to the 2 lines you chose. 6.: A= [1 2 -1 3] [1 3 2 4] [3 1 -2 4] A has rref equal to [1 0 0 1] [0 1 0 1] [0 0 1 0] a) Find a basis for R^3 from the columns of A. b) Express the last column of A as a linear combination of the other columns of A. 7.: Find the dimension of U, V and W and write down a basis for each. U= { (a,b+c-d+e,a+f,-a-f) } V= { (x,y,z,w) with 2x+y=0 } W= { all vectors in R^4 perpendicular to (1,0,-1,0) } 8.: L: R^2 -> R^2 is a linear transformation such that L((1,1)) = (1,2) and L((1,-1)) = (3,0). a) Find L((1,0)) and L((0,1)). b) Write L in standard matrix form. c) Describe the range of L. 9.: Which are subspaces? (x,y,z any real numbers) a) U= { (2x+y,y+z,2) } b) V= { (x,y,z) with x^2+z=1 } c) W= { (x,y,z) with x>0, z>0 } d) X= Span{ (1,2,3), (2,0,1) } Explain your answers. 10.: Two planes are given, E1: x+2y+2z-5=0 E2: 2x-y+z-2=0. a) Find a vector perpendicular to E1, and another vector perpendicular to E2. b) Find a point in the intersection of the planes. c) Find a parametrization of the entire intersection of the planes E1 and E2. Parts a) and b) would be useful if we had talked about vector products... 11.: A= [1 1 4 1 2] [0 1 2 1 1] [1 -1 0 0 2] [2 1 6 0 1] has rref equal to [1 0 2 0 1] [0 1 2 0 -1] [0 0 0 1 2] [0 0 0 0 0] a) Find a basis for nullspace(A). b) Are the first three columns of A linearly independent? 12.: a) Determine a basis of R^3 from among the following four vectors. v1=(1,1,0) v2=(2,1,2) v3=(3,2,2) v4=(1,0,1) -> b) Express i as a linear combination of the elements in that basis. 13.: Let V=R^4. Find an orthonormal basis for the subspace of vectors in V that are perpendicular to (1,2,3,4) and (2,3,4,5). 14.: What is the closest point to (1,2,3,4) on the subspace span((1,1,1,1),(2,4,2,6),(0,2,0,4)) ? 15.: Suppose Uli has shoesize (in inches) as follows age(years) | size 3 | 2 4 | 4 7 | 5 17 | 10 Find the best line fit for the question How big are Uli's feet after n years? According to the line fit, what is Uli's shoesize when he retires at age 72? Find also the best parabola fit for the same question. What is the prediction with this model at retirement age? Plot the given data and both models. Discuss your findings. 16.: Suppose W and W' are orthogonal complements of each other in V. Is there a relation between the dimensions of these three spaces? Prove it. 17.: Using the formula (f,g) = integral from 0 to 1 f(x)*g(x) dx compute the projection of x^2-1 to the subspace spanned by 1 and x inside P_2 (the polynomials of degree at most 2). 18.: Write (1,2,3,4) as a linear combination w+w' where w is in W = the space spanned by (1,0,0,1) and (0,1,1,0) and where w' is in the complement of W. 19.: Find an orthonormal basis for the space in R^4 consisting of all vectors like (a,a+b,b,b+c). 20.: Discuss the difference between orthogonal and orthonormal. 21.: Prove that for vectors u,v in an innerproduct space V one has 4(u,v)=||u+v||^2 - ||u-v||^2 22.: The square matrix A is called "positive definite" if for all choices of a nonzero vector v the product (v,A*v) is positive. Example: the identity matrix. But there are many others. (Note: (v,A*v)= v^T*A*v. ) Prove that if B is a nonsingular matrix then B^T * B is positive definite. 23.: Let v=(d,2,3,4). For which d is v perpendicular to the subspace of R^4 consisting of the vectors (a+b-c,a-c,0,a+b) ? How long is that v? What angle does this v make with (1,2,3,4)? 24.: Let E be the plane 3x-2y+4z=5. Find a line perpendicular to E that goes through (2,3,4). Use this line to find out how far (2,3,4) is from the line. 25.: Let L be the line through (1,2,3) with direction vector (1,-1,0), That means, the point on the line are of the form (1,2,3)+t(1,-1,0) where t is some real number. Find a line parallel to L that goes through the origin. Find a plane that is perpendicular to the line L and goes through (1,1,1). 26.: Suppose you are given a line L = (1,2,3)+t(1,-1,0) and a plane E as solutions of 3x-2y+4z=5. Where does L meet E (if at all)? 27.: Let W be the subspace in P3 of all polynomials ax^3+bx^2+cx+d for which a+b=d and 2a-c=b+c. Find a basis for W and determine its dimension. 28.: Suppose you are given vectors u=(3,4,5) and v=(1,2,3) and w=(2,3,4). Find an orthogonal basis for the space spanned by u, v and w. Does your basis involve u? Does it involve w? Comment on the answers. How would you go about finding an orthogonal basis that involves w? 29.: Consider question 14 again and see if you can come up with a better way of solving it. (Hint: look at Section 3.6)