Math 221, Fall 2007, Section 103
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Midterm Exam II October 31, 2007 No books. No notes. No calculators. No electronic devices of any kind.
Name
Student Number
Problem 1. (5 points) Give reasons why the mappings F , G and H are not linear: (a) F : R2 → R2 , defined by ! " ! " a b+1 F = b a
(b) G : R2 → R2 , defined by
(c) H : R2 → R2 , defined by
! " ! " x xy G = y x+y
! " ! " u 0 H = v |u + v|
Math 221, Midterm Exam II
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Problem 2. (4 points) Given that S : R2 → R2 is linear and that ! " ! " 1 3 S = and 2 4 find the standard matrix of S.
2
3
4
5
! " ! " 2 4 S = 3 5
6
7
total/35
Math 221, Midterm Exam II Problem 4. (6 points) The matrix
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0 0 A= 0 0
1 2 1 1
2 4 2 2
5 7 4 4
−2 −1 −1 −1
defines a linear transformation A : Ri → Rj . (a) What are the numerical values of i and j? (b) Find a basis for the image space W ⊂ Rj . (c) What is the dimension of the image space W ? (d) What is the dimension of the kernel of the linear trasnformation A?
Math 221, Midterm Exam II Problem 6. (6 points) It is given that
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! " ! " 4 −3 ) B=( , 4 3
is a basis of R2 . Please note that the two vectors in B are orthogonal to each other. Denote the standard basis of R2 by E. Let T : R2 → R2 be orthogonal projection onto the line ! " −3 L = span 4 (a) (b) (c) (d)
Find Find Find Find
the the the the
matrix [T ]B of T in the basis B. matrix P such that [!v ]E = P [!v ]B , for all vectors !v ∈ R2 . matrix Q such that [!v ]B = Q [!v ]E , for all vectors !v ∈ R2 . standard matrix [T ]E of the projection T .
Math 221, Midterm Exam II
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Problem 7. (5 points) Find the standard matrices of the following linear transformations from R2 to R2 . (a) R: rotation by 60◦ . (b) P : orthogonal projection onto the line x + y = 0. (c) S: reflection across the y-axis. (d) P 2 (e) R ◦ S
Math 221, Section 103, Final Exam
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Problem 10. (8 points) A discrete dynamical system is defined by xn+1 = xn + 3yn yn+1 = 2xn + 2yn
x0 = 5 y0 = 10
(a) (6 points) Find explicit formulas for xn and yn , with the given initial condition. xn+1 (b) (1 point) Find the limiting growth rate of x, i.e., find lim . n→∞ xn xn (c) (1 point) Find the limit lim . n→∞ yn
Math 221, Section 103, Final Exam
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Problem 9. (8 points) On a remote planet, moisture is present in clouds, on the continents, and in the seas. Each year 80% of the cloud moisture falls onto the land and 10% falls into the seas. Each year 15% of the land moisture evaporates directly into the clouds and 65% runs into the seas. Each year 70% of the sea moisture evaporates into the clouds. Assume we know also that the total amount of water or moisture on the planet is 86 trillion litres. (a) Draw a diagram of the moisture transfer on this planet. (1 point) (b) Write a system of linear equations describing the moisture transfer using the variables c, l, and s for water content in clouds, on land, and in the seas, respectively. Find the transition matrix of this dynamical system. (2 points) (c) Suppose the planets moisture distribution is in equilibrium. What is the annual precipitation (volume of water falling onto land from the clouds)? (3 points) (d) A large meteor falls onto the planet, causing all sea water to evaporate into the clouds. (The impact has no other effect on the moisture distribution, but it does cause mass extinction of aquatic species.) ! c " Write down the matrix which represents the change in the state vector sl caused by the impact. (2 points)
Math 221, Section 103, Final Exam
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Problem 8. (8 points) For each of the linear maps T : R2 → R2 do the following: find a basis of R2 consisting of eigenvectors of T , or explain why this is not possible. (a) T : R2 → R2 is given by reflection across the line y = 12 x. (b) T : R2 → R2 is given by rotation clockwise 5 degrees. (c) the dilation T : R2 → R2 given by T (!v ) = 5!v , for all !v ∈ R2 . (d) the shear in the y-direction whose matrix is ! " 1 0 2 1 3
Math 221, Section 103, Final Exam
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Problem 6. (6 points) ! " ! " 1 1 (a) Explain why B = { , } forms a basis for R2 . 1 −1 ! " 7 (b) Find the coordinate vector of in the basis B. 10 (c) Suppose the standard matrix of a linear transformation T : R2 → R2 is ! " 2 −3 0 2 Find the matrix of T with respect to the basis B, i.e., find [T ]B .
Math 221, Section 103, Final Exam Problem 3. (6 points) Find a 2 × 2 matrix A such that ! " ! " ! " 1 2 1 3 1 1 A = . 0 1 0 1 1 1
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Problem 4. (9 points) Consider the following three linear transformations from R2 to R2 . ! " 1 −2 The shear S = , 0 1 ! " 0 1 the reflection R = , 1 0 ! " √ 1 −1 2 the projection P = 2 . −1 1 (a) Which ones of these linear transformations are invertible?
(b) Describe geometrically the null space and the range of each of S, R, P . N.B. The range of a linear transformation is equal to the column space of its matrix.
Problem 4. (continued) (c) Find the matrices of the following three transformations: (i) the composed transformation obtained by first shearing and then reflecting, i.e., first applying S and then R..
(ii) the composed transformation obtained by first reflecting and then shearing.
(iii) the composition of P with itself.