Math 3130
Spring 2016 Abrams
Practice Exam for Exam 2
Show all your work. No credit will be given for answers which are not accompanied by supporting computations. Use the back of the sheet if you need more space. A non-graphing calculator with no QWERTY keyboard
is allowed, but definitely not needed. Circle answers when appropriate. Good luck.
1. (6 pt total) Linear Transformations.
(a) Define precisely: If T : V → W is a function from the vector space V to the vector space W ,
the T is called a linear transformation in case:
(i)
and
(ii)
(b) Use the definition you gave in part (a) to show that the function T : R3 → R3 given by the formula
T ((x, y, z)) = (x + y, y, z + 2x) is a linear transformation.
2. (1 pt each) Circle the functions T which are linear transformations.
(i) Let A be an m × n matrix. Define T : Rn → Rm by T (v) =Av for each column vector v in Rn .
(ii) T : R2 → R2 where T rotates each vector in R2 π/4 radians clockwise around the origin.
(iii) T : P3 → P2 where T (p(x)) = p0 (x).
(iv) T : P2 → P2 where T (a0 + a1 x + a2 x2 ) = a0 + a1 (x + 3) + a2 (x + 3)2 .
3. (3 pt) If B = {1, x, x2 } is the standard basis for P2 , and T : P2 → P2 is a linear transformation with the
property that T (1) = x, T (x) = 3x2 + 1, and T (x2 ) = 1 + 2x, compute T (4 − x + 2x2 ).
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4. (3 pt total) (i) Let V be an n-dimensional vector space with basis B = {u1 , u2 , ..., un } and let W be
an m-dimensional vector space with basis B 0 = {w1 , w2 , ..., wm }. Let T : V → W be any linear transformation.
Let A denote the matrix
A=
(ii) (Do just ONE of these.)
Then for every vector v in V,
A[v]B =
OR
T (v) =
(iii) Draw a diagram (containing four vector spaces, and four arrows) which presents the information
given in parts (i) and (ii). Make sure to clearly include and label all appropriate sets, functions, vector spaces,
arrows, etc ...
5. (8 pt total) Define T : P3 → P3 by setting T (p(x)) = 4p0 (x) − 2p(x) for every p(x) ∈ P3 . (So, for example,
T (x2 ) = 8x − 2x2 .). It is not hard to show that T is a linear transformation. (You need not show it here, but
you may assume it’s true.)
(a) Let B = {1, x, x2 , x3 } be the standard basis for P3 . Find the matrix of T with respect to B. Call
this matrix A.
(b) Let q(x) = 2 − x2 + x3 . Find the coordinate vector [q(x)]B .
(c) Compute the matrix product A[q(x)]B .
(d) Compute T (q(x)).
(e) Find the coordinate vector [T (q(x))]B .
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6. (12 pt total) Let T : R2 → R2 be the linear transformation defined by T ((x, y)) = (x + 7y, 3x − 3y).
(a) Find the matrix [T ]B of T with respect to the standard basis B = {e1 , e2 } of R2 . Call this matrix A.
(b) Let C be the basis C = {(1, 1), (0, 1)} of R2 .
this matrix M .
Find the matrix [T ]C of T with respect to C.
Call
(c) Let P denote the transition matrix from C to B. Find P.
(d) Let Q denote the transition matrix from B to C.
Find Q.
(e) Give an equation which relates the four matrices A, M, P, and Q. (Your answer will contain these
four letters.)
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7. (3 pt) Let T : R2 → R2 be the linear transformation ’rotate through 45 degrees counterclockwise’. Find
the matrix of T with respect to the standard basis of R2 .
8. (2 pt) Let T : R3 → R3 be the projection of R3 into the xz-plane. (So T ((x, y, z)) = (x, 0, z).) Find a
basis of the kernel of T .
9. (1 pt each) True - False
(a)
T F Suppose V and W are finite dimensional vector spaces. By appropriately using coordinate
vectors, it is possible to view any linear transformation T : V → W as a matrix multiplication.
(b)
T F Suppose T : V → W is a linear transformation, and suppose B = {b1 , b2 , ..., bn } is a
basis of V . If you are told the values of T (b1 ), T (b2 ), ..., T (bn ), then you have enough information to compute
T (v) for every v in V .
(c)
T F It is possible to have an n × n matrix A with the property that for every invertible n × n
matrix P we have A = P −1 AP.
(d)
T F If V is a vector space and k is any scalar, then the function T : V → V given by defining
T (v) = kv for every v ∈ V is a linear transformation.
(e)
T F If T : P∞ → P∞ is the linear transformation given by T (f (x)) = f 0 (x)
’derivative’ linear transformation on polynomials), then the kernel of T is {0}.
(i.e., T is the
(f)
T F If T : R∞ → R∞ is the linear transformation given by T ((a0 , a1 , a2 , . . . )) = (0, a0 , a1 , a2 , . . . )
(i.e., T is the “right shift” linear transformation), then the kernel of T is the zero vector of R∞ .
(g)
T
F If M, A, and P are n × n matrices for which M = P −1 AP , then det(M ) = det(A).
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