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(MATH113)midterm_sol_yrXXXX.pdf
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Midterm Soltions
1.For what values of h the following system of linear equations has(1)unique,(2)in.nitely many solu-
tions,(3)no solution.(Each part values 10 points.)
Answers:
8 ... ..:
2
x1+x2+x3 . 1
2x1+x2+ hx3 . 2
.2x1+ hx2. hx3 . h.4
3
2
3
1111 111 1
5
4
5
4
21h 2 .0.1 h.2 0
.2 h .h h.4 0 0 (h. 2)(h+1) h.2
(1)
unique solution i. h 6. .1; 2;
(2)
in.nitely many solutions i. h . 2;
(3)
no solution i. h . .1.
44
2. Let T : R ! R be a linear transformation de.ned by
T(x1;x2;x3;x4) . (x1 +x2 .x3 .x4;.x1 +x2 .x3 +x4;x1 +x2 .2x3;x1 .x2 .3x3 +2x4):
Is T invertible.If yes,.nd its inverse linear transformation(20 pts).
Answers:
Yes.
.1
T (x1;x2;x3;x4) . (.7.2x1.3.2x2+4x3.x4;.5.2x1.1.2x2+3x3.x4;.3x1.x2+3x3.x4;.4x1.
x2 +4x3 .x4):
33 44 34 43
3. LetT1:R !R;T2:R !R;T3:R !R;T4:R !R;be linear transformations de.ned
by
T1(x1;x2;x3) . (.x1 +x2 +x3;x1 .x2 +x3;x1 +x2 .x3):
T2(x1; x2; x3; x4) . (x1+2x2+3x3+4x4; 2x1+3x2+4x3+5x4; 3x1+4x2+5x3+x6; 4x1+5x2+63x3+7x4):
T3(x1; x2; x3) . (x1 + 2x2 + 3x3; 2x1 + 3x2 + 4x3; 3x1 + 4x2 + 5x3; 3x1 + 2x2 + x3):
T4(x1;x2;x3;x4) . (x1 +x2 +x3 +x4;x1 +x2 +2x3 +2x4;x1 +2x2 +x3 +2x4):
respectively.Which one of them is (a)one-to-one,but not onto;(b)onto,but not one-to-one;(c)both one-to-
one and onto,and (d)neither one-to-one nor onto.Please gives brief reason(40 pts).
Answers:
3
2
3
2
3
2
3
2
.11 1 .111
1234 1234
664
2345
775
.
664
0 .1 .2 .3
775
;
5
4
5
4
A1. 1.11 . 020;A2.
3456 0000
1 1.1 002
4567 0000
3
2
3
2
123 123
3
2
3
2
1111 1111
A3 .
664
234
775
.
664
0 .1 .2
775
;A4 .
5
4
5
4
1122 . 0101
:
345 000
1212 0011
321 000
1.
No one is (a)one-to-to,but not onto;
2.
A4 is (b)onto,but not one-to-one;
3.
A1 is (c)both one-to-one and onto;
4.
A2 and A3 are neither one-to-one nor onto.
4. Circle only YES or NO;no reason needed(20 pts).
nm
Let A be an mxn matrix and T : R ! R a linear transformation de.ned by T (x) . Ax:
(a)
If the column vectors of A are indep endent, then T is one-to-one.
(b)
If the row vectors of A are indep endent, then T is one-to-one.
(c)
If the column vectors of A are indep endent, then T is onto.
(d)
If the row vectors of A are indep endent, then T is onto.
(e)
If every row of A has a pivot position, then T is one-to-one.
(f)
If every column of A has a pivot position,, then T is one-to-one.
(g)
If every row of A has a pivot position, then T is onto.
(h)
If every column of A has a pivot position,, then T is onto.
(i)
If v1;... ;vk are linearly independent vectors in R , then T(v1);... ;T (vk) are linearly independent.
(j)
If v1;... ;vk are linearly dependent vectors in R , then T(v1);... ;T (vk) are linearly dependent.
(a)
Yes (b) No (c) No (d)Yes (e)No (f )Yes (g)Yes (h)No (i)No (j