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Math113 L1: Linear Algebra Midterm Exam (a) Dept of Math, HKUST, Spring 2004
Name: Tutor:
ID No. Section:

Problem No.1 (18 pts) No.2 (15 pts) No.3 (13 pts) No.4 (24 pts) Total (70 pts)
Score

1. (15 pts) Let T : R3 R3 de.ned by T (x1,x2,x3)=(x2 +2x3,x1 +2x2, 2x1 + x3),
(a)
(3 pts) Find the standard matrix of T .

(b)
(5+10 pts) Is T invertible? If so, .nd the standard matrix of its inverse linear transformation.


1

2. (15 pts) Let T : R4 R4 be a linear transformation whose standard matrix is
1 1 2 2
2 2 3 3
3 3 4 4
4 4 5 5

.

..
.
..
.
(a)
(10 pts) Find the basic solutions for the homogeneous system T (x1,x2,x3,x4) = (0, 0, 0, 0).

(b)
(5 pts) Find all solutions for the non-homogeneous system T (x1,x2,x3,x4) = (2, 3, 4, 5).


2

3. (a) (8 pts) Let T1 : R2 R2 be the rotation about 7 counterclockwise. Find the standard matrix of T1.
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(b) (5 pts) Let T2 : R2 R2 be the re.ection about the straight line x . y = 0. Find the standard matrix of T2.
4. (24 pts)
(a)
Let T : Rn Rm be a linear transformation, and let v1,... ,vk be vectors of Rn .

(i)
If v1,... ,vk are linearly dependent, then T (v1),... ,T (vk) are linearly dependent. YES or NO

(ii)
If v1,... ,vk are linearly independent, then T (v1),... ,T (vk) are linearly independent. YES or NO


(iii) If T (v1),... ,T (vk) are linearly dependent, then v1,... ,vk are linearly dependent. YES or NO
(iv) If T (v1),... ,T (vk) are linearly independent, then v1,... ,vk are linearly independent. YES or NO

(c)
Let T : Rn Rm be a linear transformation having the standard matrix A.

(i)
T is one-to-one if and only if the number of pivot positions of A is equal to n. YES or NO

(ii)
T is one-to-one if and only if the number of pivot positions of A is equal to m. YES or NO




(b) (i) Let v1, v2, v3, v4 be vectors of R4 of and v4 = 0. Are v1, v2, v3, v4 dependent? YES or NO
(ii) Is it possible to .nd more than eight linearly independent vectors in R5? YES or NO
(iii) If v1, v2, v3, v4 are independent, then v1, v2, v3 are independent. YES or NO
(iv) If v1, v2, v3, v4 are dependent, then v1, v2, v3 are dependent. YES or NO

(iii) T is onto if and only if the number of pivot positions of A is equal to m. YES or NO
(iv) T is onto if and only if the number of pivot positions of A is equal to n. YES or NO
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