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242 MA More on linear transformations Basic facts about linear transformations T 1. T (0) = 0 2. T (r1 v1 + . . . + rk vk ) = r1 T (v1 ) + . . . + rk T (vk ) September 24, 2007 Last class we discussed the following list of functions. (Some are linear, but others are not.) Examples: functions f : R R 1. f1 (x) = 2x 2. f2 (x) = 2x + 1 3. f3 (x) = x2 4. f4 (x) = cos x Examples: functions g : R2 R2 1. g1 (x1 , x2 ) = (x1 + x2 , 2x1 x2 ) 2. g2 (x1 , x2 ) = (cos(x1 + x2 ), x1 + x2 ) 2 Examples: functions h de ned on R3 1. h1 (x1 , x2 , x3 ) = (x1 + x3 , x1 x2 + x3 ) 2 1 1 2. h2 (x1 , x2 , x3 ) = 1 2 3 1 1 x1 1 1 x2 x3 2 One way to show that a transformation is linear is to verify the two conditions of linearity directly, but there is an easier way to see that these transformations are linear. Important class of examples: Given an m n matrix A, then we can de ne a linear transformation T : Rn Rm by the equation T (x) = Ax. We know that T is a linear transformation because the matrix-vector product satis es the necessary conditions. 1 MA 242 Example. Let G= Then G x1 x2 = 1 1 2 1 . September 24, 2007 On the course web page, there is a link to a java program called Matrix Machine written by Hu Hohn. It lets you investigate the mapping properties of various 2 2 matrix transformations. Try the following three matrices: 2 0 0 3 2 1 1 1 0 1 1 0 There also are links to two applets by David Austin at Grand Valley State University. These two applets are particularly useful when you want to understand the mapping properties of linear transformations from R2 to R2 . Theorem. Every linear transformation T : Rn Rm can be written as T (x) = Ax, where A is an m n matrix. This matrix A is called the standard matrix representation of T . Why? Let s make two observations using the standard basis of Rn . De nition. The vectors e1 , . . . , en in Rn are the vectors e1 = 0 1 0 . . . 0 e2 = 0 0 1 . . . 0 ... en = 0 0 0 . . . 1 . 2 MA 242 September 24, 2007 1. If we know the images T (e1 ), . . . , T (en ) for all of the standard basis vectors e1 , . . . , en , then we can calculate T (x) for any x in Rn . 2. If A = T (e1 ) T (e2 ) then ... T (en ) , Ax = x1 T (e1 ) + . . . + xn T (en ). 3 MA 242 September 24, 2007 Example. What is the matrix representation of the linear transformation that rotates R2 by 45 ( /4 radians) around the origin? 4 MA 242 Examples of linear transformations T : R2 R2 1. rotations September 24, 2007 2. re ections 3. contractions and expansions 4. shears 5. projections 5
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BU >> MATH >> 242 (Fall, 2009)
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BU >> MATH >> 225 (Fall, 2009)
MA 225 More on triple integrals Last class we spent quite a bit of time discussing the following example. Example. Evaluate z dV Q November 10, 2008 where Q is the region bounded by the cylinder x2 + z 2 = 9, the plane y + z = 3, and the plane y = ...
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MA 242 Elementary row operations on a matrix September 10, 2007 1. (Replacement) Replace one row by the sum of itself and a multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Replace a row by a nonzero multiple of itself. ...
BU >> MATH >> 231 (Fall, 2009)
MA 231 September 20, 2002 Homework Problems on Picard Iteration 1. Calculate the rst three Picard iterates y0 (t), y1 (t), and y2 (t) for the following initialvalue problems: a. dy/dt = y 2 1, y(1) = 2 b. dy/dt = t2 + y 2 , y(0) = 1 c. dy/dt = et ...
BU >> MATH >> 242 (Fall, 2009)
MA 242 Exams Fall 2006 Here are the four exams that I gave in my MA 242 class in the Fall of 2006. You should use them to get an idea of the format of a typical test and to see the types of questions I ask. You should not assume that the test questio...
BU >> MATH >> 242 (Fall, 2009)
MA 242 September 19, 2005 Given the matrix-vector product Ax, we can now establish the following theorem. Theorem. Let A be an m n matrix. Then the following three statements are equivalent: 1. For each b in Rm , the equation Ax = b has at least o...
BU >> PY >> 105 (Fall, 2009)
Physics Workshop Centripetal Motion Suggested Checkpoint Questions with Answers Checkpoint following 1(c): 1. What is the direction of the net force in each of your free-body diagrams? How do you know? 2. Which force or forces are supplying the net...
BU >> PY >> 211 (Fall, 2009)
Challenge Problem Set 1 Solutions PY 211 C. A. Serino February 2007 1.1) A string of total mass, M, and length, , is situated on a table such that, initially, a small amount, , hangs over the edge as shown in Figure 1. At what time, t, does the stri...
BU >> PY >> 211 (Fall, 2009)
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BU >> PY >> 211 (Fall, 2009)
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BU >> INTRO >> 2 (Fall, 2009)
GEOMETRICAL OPTICS Geometrical optics is the treatment of the passage of light through lenses, prisms, etc. by representing the light as rays. A light ray from a source goes in a straight line through the air, but when it encounters a lens, prism, or...
BU >> PY >> 212 (Fall, 2009)
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BU >> PY >> 211 (Fall, 2009)
PY 211 Mid Term Exam II April 3, 2007 Formula Sheet, Useful Data, and Moments of Inertia Formula Sheet, Useful Data, and Moments of Inertia (Cont.) Useful Numbers: G = 6.67 x 10-11 Nm2/kg2 g = 9.8 m/s2 REarth = 6.38 x 103 km MEarth = 5.97 x 1024 ...
BU >> PY >> 105 (Fall, 2009)
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BU >> LX >> 502 (Fall, 2009)
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BU >> MATH >> 242 (Fall, 2009)
MA 242 More on bases October 26, 2007 We need ways of determining bases of vector spaces and their subspaces. The casting-out procedure produces a basis from a spanning set. The casting-out procedure Given a vector subspace S spanned by {v1 , v2 , ...
BU >> MATH >> 226 (Fall, 2009)
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BU >> MATH >> 226 (Fall, 2009)
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BU >> MATH >> 242 (Fall, 2009)
MA 242 More matrix algebra Denition. The product C of A and B is the matrix C = AB1 September 29, 2008 AB2 . ABp where B1 , B2 , . . . , Bp are the columns of the matrix B. Example. Let A : R2 R2 be rotation by 45 and let B : R2 R2 be t...
BU >> MA >> 225 (Fall, 2009)
Math 225 Section A1 Spring 2005 M, W, F, 9:00am 10:00am, STO B50 Instructor: Russell K. Jackson office: 111 Cummington St, MCS 236 phone: (617) 353 - 1487 e-mail: russjack@bu.edu office hours: Mondays - 1:15pm - 2:45pm Wednesdays - 10:15am - 11:45am...
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