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- Title: midterm_S04
- Type: Notes
- School: Princeton
- Course: MAT 202
- Term: Spring
Coursehero >> New Jersey >> Princeton >> MAT 202
Path: Princeton >> MAT >> 202 Spring, 2008
Path: Princeton >> MAT >> 202 Spring, 2008
Path: Princeton >> MAT >> 202 Spring, 2008
Path: Princeton >> MAT >> 202 Spring, 2008
Path: Princeton >> MAT >> 202 Spring, 2008
Path: Princeton >> MAT >> 202 Spring, 2008
Path: Princeton >> MAT >> 202 Spring, 2008
Path: Ball State >> PEP >> 291 Spring, 2008
Path: Ball State >> PEP >> 291 Spring, 2008
Path: USC >> BISC >> 330L Spring, 2006
Path: Ball State >> PEP >> 291 Spring, 2008
Path: USC >> BISC >> 330L Spring, 2006
Path: Ball State >> PEP >> 291 Spring, 2008
Path: USC >> BISC >> 330L Spring, 2006
Path: Ball State >> PEP >> 291 Spring, 2008
Path: USC >> BISC >> 330L Spring, 2006
Path: Ball State >> PEP >> 291 Spring, 2008
Path: USC >> BISC >> 330L Spring, 2006
Path: Ball State >> PEP >> 222 Spring, 2008
Path: Ball State >> PEP >> 222 Spring, 2008
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Path: Kettering >> CHEM >> 135 Summer, 2005
Path: Kettering >> CHEM >> 135 Summer, 2005
Path: Kettering >> PHYS >> 114 Winter, 2008
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Path: Hofstra >> ENGG >> 10 Fall, 2006
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Path: Texas A&M >> PHYS >> 222 Spring, 2008
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Path: Texas A&M >> MATH >> 311 Spring, 2008
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Instructor Name Time your class meets MATH 202 - MIDTERM EXAM Wednesday, March 10, 2004, 7:30PM-9:00PM McCosh 10 Average was 51 1. (16 points) The linear transformation T transforms vectors in R2 in the following way: first it rotates the vector through an angle of 30 degrees counterclockwise about the origin, then it doubles the vector's length and then it reflects the resulting vector across the line of slope 1 through the origin. What is the standard matrix of T ? 2. (16 points) Let A = 1 2 0 0 0 1 -1 -2 1 2 4 -1 2 3 1 1 which row reduces to rref (A) = 1 0 0 0 2 0 0 0 0 1 0 0 2 3 0 0 . (a) Find a basis for the image of A. (b) Find a basis for the kernel of A. (c) Find a matrix B such that the image of B is the same as the kernel of A. (d) Find a matrix C such that the kernel of C is the same as the image of A. 3. (16 points) Let V be the subspace of R4 spanned by the vectors v1 = (a) Find a basis for V . 1 1 1 1 v2 = 1 2 -1 0 v3 = 0 1 1 -1 (b) Find the matrix for orthogonal projection onto V . (c) Use your result from part (b) to find the matrix of orthogonal projection onto V . 4. (16 points) Let T : R2 R2 be a linear transformation such that T (T (x)) = 0 for every x in R2 and also suppose that there is at least one vector x0 where T (x0 ) = 0. Show (a) that x0 and T (x0 ) are linearly independent. (b) Let B be the basis {x0 , T (x0 )} of R2 . What is the matrix of T with respect to this basis? (c) What is the square of the matrix you found in part (b)? (d) If T is such a transformation and T the standard basis? Note: You don't need to be able to do part (a) in order to do the other parts of this question. In fact all four parts can be figured out independently from each other. 1 2 = 6 3 then what is the matrix of T with respect to 5. (16 points) Consider the rank 3 matrix A= 1 2 3 0 1 -3 1 -2 -3 1 0 -3 Find an orthonormal basis for the image of A. 6. (20 points) True or False. If true, explain why. If false, give a counter-example. (a) Let P be any plane through the origin in R3 . Let A be the matrix for reflection across the plane 1 0 0 P (with respect to the standard basis). Then A is similar to 0 -1 0 . 0 0 1 (b) Let V be the subset of R4 defined by the equation x1 + x2 + x3 - x1 x4 = 0. Then V is a linear subspace of R4 . (c) If V and W are subspaces of Rn then their union V W (defined to be the set of all vectors x in Rn such that x is in V OR x is in W ) is also a subspace of Rn . (d) The matrix -34 25 -49 36 is a shear. (e) If {v1 , v2 , v3 } is a linearly independent set of vectors in Rn and S is any invertible n n matrix then {Sv1 , Sv2 , Sv3 } is also linearly independent.
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PHYS-114 [Russell] Test 3 (Winter 08)Path: Kettering >> PHYS >> 114 Winter, 2008
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