Review Test 2 - Chapter 4+5+6

MAA101 - Review Chapter 4+5+6
1. Find all real numbers x so that [1 -2 5]T and [2 3 x]T are orthogonal.
A) X = 4/5

B) x= 2/3

C) x = 0

2. Find the area of the triangle with vertices:

27
4

A)

27
2

B)

D) None

A(3, 1, -2); B(5, 2, -1);
C)

27

C(4, 3, -3)

D) None

3. Find a parameter equation of the line consisting of all points of intersection of the two planes:
(P): x- y + z=2; (Q): 2x-3y+4z=7
 x  1  2t
A)  y  3  2t
 z  4t


 x  7  2t
B)  y  13  4t
 z  2t


 x  1  t
C)  y  3  2t
 zt


D) None

4. Find the length of the projection of vector u(1, 2, 4) on vector v(1, -2, 1).
A) (1/6; -1/3; 1/3)

B) 1/ 6

C) 6

D) None

5. Find the distance from the origin to the line passing through the point (3, 1, 5) and in the direction
of the vector (2, 1, 1).

A)

11

B)

5 2
3

C)

6
3

D) None

6. Find the distance from P(1, -2, 3) to the plane 2x-y-z=6, and the point H on the plane that is closest
to the that plane.

1
14  10 13
;H( ,
, )
6 6 6
6
C) d  5 ; H ( 1 ,  13 , 21)
6 6 6
6

A) d 

B)

d

5
16  17 13
;H( ,
, )
6 6 6
6

D) None of the other choices is true


7. Find a parameter equation of the line passing through P(1, 1, 1) and perpendicular to the lines

[ x, y, z ]  [1,1,0]  t[2,0,1] and[ x, y, z ]  [2,1,3]  t[0,0,1]
A)[x, y, z ]  [1,1,1]  t[0,1,1] B)[x, y, z ]  [1,1,1]  t[0,1,0]
C )[x, y, z ]  [1,1,1]  t[2,1,1] D) None
8. Let T: R2→R2; T(x,y) = 1/5(-3x+4y, 4x+3y). Which of the following statements are true?
A) T is projection on a line.
B) T is reflection in a line
C) T is rotation through an angle.
D) T is not a matrix transformation.
9. Let T: R2→R2,

T ( x, y ) 

1
( x  y; x  y ).
2

Which of the following statements are true?

A) T is projection on a line.
B) T is reflection in a line
C) T is rotation through an angle.
D) T is not a matrix transformation.
10. Find the (2,3)-entry of the matrix of projection on the line [x, y, z]= t[1,1,0].
A) 0

B) 1

C) ½

D) None

11. Find the (3,2)-entry of the matrix of reflection in the plane x + 2y - z=0
A) 0

B) 1/3

C) 2/3

D) None

12. Which of the followings are subspaces?
(i)

U = {(x, y, z) Є R3 I x-4y+5z = 0}

(ii)

U = {p(x) Є P2 I p(0) + P(5) = 0 }

(iii)

U = {A Є M22 I A is invertible}

A) All of them

B) (i)

C) (i) and (ii)

D) None


13. Which of the following statements are true?
(i)

P2 = span{1+x; -x-1; 3x2}

(ii)

(2, 3, 1) is a linear combination of (1, 1, 1) and (0, 1, -1).

A) Both

B) (i)

C) neither

D) (ii)

14. Which of the following statements are true?
(i) If {u, v} is a spanning set of a subspace U then {u, v, w} is also a spanning set of U for any w Є U.
(ii) If {u, v, w} is independent then {u, v} is dependent.
(iii) Every set of vectors that contains the zero-vector is dependent.
(iv) The set of vectors {u, 2u, v, w} is independent.
15. Which of the following statements are independent?
(i)

{(1, 2, 3); (2, 0, 0); (3, 1, 1)}

(ii)

{p1(x) = x2+x-2; p2(x) = 2x - 1 }

(iii)

1 0 0 4 0 0 

; 0 5; 0 1 
2
0






A) All of them

B) (i)

C) (i) and (ii)

16. Find a basis and calculate the dimension of the subspace
U = span{(1, 0, 3); (2, 1, 3); (1, 1, 1)}
A) Dim = 3; basis = {(1, 0, 3); (2, 1, 3)}
B) Dim =2; basis = {(1, 0, 3); (2, 1, 3)}
C) Dim = 3; basis = {(1, 0, 3); (0, 1, -3); (0, 0, 1)}
D) Dim = 1; basis = {(2, 1, 3)}.

D) None


17. Find a basis and calculate the dimension of the following subspace.
U = {p(x) Є P3 : p(4) = P(0) = 0}
A) Dim = 1; basis = {x3 - 64}
B) Dim = 2; basis = {x3-16x; x2-4x}
C) Dim = 3; basis = {x3-16x; x2-4x; -x3+16x}
D) None of the other choices is true
18. Calculate the dimension of the following subspace.
A) 1

B) 2

C) 3


1 0
0 0  
X 
 X  M 22 : 


1 0
0 0  


D) 4

1 4 0
19. Let λ = 1 be an eigenvalue of A = 

0 5 0 
corresponding to λ = 1.

. Find the dimension of the eigenspace of A

0 0 2 

A) 1

B) 2

C) 3

D) 0

20. Let A be a 20x60 matrix so that rank(A) = 10. Find Dim(Col(A)), dim(Row(A)), dim(Null(A)).
A) Dim(Col(A)) = 60, dim(Row(A))=20, dim(Null(A))=10.
B) Dim(Col(A)) = 10, dim(Row(A)) =20, dim(Null(A)) =40.
C) Dim(Col(A)) = 10, dim(Row(A)) = 10, dim(Null(A)) = 50.


Key: 1A 2B 3C 4B 5A 6B 7B 8B 9C 10A 11C 12C 13D 14(i),(iii) 15A 16C 17B 18B 19A 20C