Download Math 342 Homework Due Tuesday, April 6 1. Let B be the basis of R

Math 342
Homework Due Tuesday, April 6
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1. Let B be the basis of R3 consisiting of the vectors  2 , 0, and 1. Find PB←E . For each of the
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following vectors v, find [v]B .
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(a) v = 3 ,
(b) v = 1 ,
(c) v = 0 .
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2
denote the polynomial t(t−1)(t−2)···(t−k+1)
. For instance, 2t is the polynomial t2 − 2t and 0t = 1.
k!
The polynomials 0t , 1t , 2t , and 3t form a basis B for P3 . Find the coordinates of t2 and t3 with
respect to this basis—i.e., determine [t2 ]B and [t3 ]B .
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3. Someone is working with a basis B for R and tells you that the change-of-basis matrix PB←E is
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2 5
What is the basis B?
3 7
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4. Someone is working with a basis B for R and tells you that the change-of-basis matrix PE←B is
.
1 −2
What is the basis B?
2. Let
t
k
5. Find the equation (in x, y-coordinates) for the ellipse whose major axis is along the line y = 3x, which
intercepts this line a distance of 8 units from the origin, and which intercepts the line y = − 31 x a distance
of 1 unit from the origin.
6. Rotate the parabola y = x2 clockwise until its axis of symmetry is along the line y = x. Find the equation
in x, y-coordinates for this new parabola. (Hint: Start by writing down a new basis B, consisting of
perpendicular unit vectors, where it’s easy to give the equation for the parabola).
7. Let B denote the basis
{1, t, t2 , t3} for P3 , and suppose T : P3 → R2 is the linear transformation for
3 2 1 0
which ME←B (T ) =
. Determine T (1 + 3t) and T (−2 − t + t3 ).
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)
(
a b 8. Let V =
a − 3b + c − 2d = 0 and 2a + b + c − d = 0 , which is a subspace of M2×2 . Find
c d a basis for V . (Hint: Use an isomorphism M2×2 → R4 to translate this into a problem about R4 ; solve
the problem, then translate back.)