Linear Algebra questions

  

Week 4 questions.docx
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1: Given a linear transformation
T : R2 ? R2
defined by
? ? x1 ? ? ? 2 x1 – x2 ?
T ?? ? ? ?? = ?
?
x
x
+
x
?
?
?
1
2 ?
? 2 ?
a) Find the matrix of this linear transformation with respect to the
standard ordered bases B to B, where B=
? ?1 ? ? 0? ?
? ? ?, ? ? ?
? ? 0? ?1 ? ?
b) Find the matrix of this linear transformation with respect to the
ordered
? ?1? ? 1 ? ?
bases C to C, where C= ?
?1? , ? – 1? ?
?? ? ? ? ?
2. For the linear transformation
(
)
D : P2 ? P2 defined by
D a0 + a1 x + a2 x 2 = a1 + 2a2 x
a) Find the matrix of this linear transformation with respect to the
standard ordered bases B to B, where B=
?1, x, x 2 ?
b) Find the matrix of this linear transformation with respect to the
ordered bases C to C, where C=
?1 – x,1 + x, x 2 ?
T : R 2 ? P1 be the linear transformation defined by
? ?1 ? ? 0? ?
? 2 1?
with respect to the bases ?
A= ?
, ? ? ? and ?1 – x,1 + x?
?
?
?
?1 1?
? ? 0? ?1 ? ?
3. Let
of
R
2
and
?? 3 ??
P1 respectively. Find T ?? ? ? ?? .
? ? – 2? ?
Solve this problem using this example: 2.1 Example Consider this matrix.
1 2
H =(3 4)
5 6
It is 3×2 so any map that it defines must carry a dimension 2 domain to a
dimension 3 codomain. We can choose the domain and codomain to be
R2 and?2 , with these bases
1
1
? = <( ) , ( )> D =
1
-1
Then let h: R2 -> ?2 be the function defined by H. We will compute the image
under h of this member of the domain.
-3
(?? = ( ))
2
The computation is straightforward.
-11/2
1 2 -1/2
???? (h(??)) = H · ???? (??) =(3 4) (
) = (-23/2)
-5/2
-35/2
5 6
From its representation,
computation of ?
??? is routine (-11/2)(? 2 )-(23/2)(? 2 + ?) – (35/2)(? 2 + x + 1) =
(-69/2)? 2 – (58/2)x – (35/2).
6. Use the method of example 4.9 to find the inverse of the matrix
? 2 1 2?
? 1 1 3?
?
?
?? 2 1 1 ??
4.9 Example This one happens to start with a row swap.
0
(1
1
3 -1 1 0
0
1 |0 1
-1 0 0 0
0
1 0
1 0
?1 < -> ?2 (0 3 -1|1
0) ?????????????????????????
1
1 -1 0 0
1
???????????????????
-?1 + ?3 (0
0
0
1 0
3 -1|1
-1 -1 0
1 0
0 0)
-1 1
?
1 0
? (0 1
0 0
0 1/4
0| 1/4
1 -1/4
1/4 3/4
1/4 -1/4)
3/4 -3/4
1 0
0 0)
0 1

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