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(math111)[2009](f)midterm~ma_yxf^_10435.pdf
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Midterm Exam for Math 111 (L1), Oct. 22, 2009

Name: ................., ID: ................., Score: .................
There are 25 questions, some of which have two or more correct statements. Please tick (.) all correct statements. (Note that the questions are printed on both sides of this sheet.)
(1) Equation
2
a) 2x +3xy = 0, .b) 2x +3y =0, c)2x . 3y = 1, d) x+ y2 =1
is a homogeneous linear equation.

(2)
Equation 2x +3y = 1 represents a line in R2 passing through point
a) (0, 0), b) (2, .3), c) (3, .2), .d) (.1, 1).


(3)
Linear system


. . 2x +3y =1
6x +9y =3
.
4x +6y =2
has
a) no solution, b) unique solution, .c) in.nitely many solution.

(4) Vector equation
.. ....
2 31 x .3. + y .9. = .2.
4 63
has
.a) no solution, b) unique solution, c) in.nitely many solution.
(5) Matrix equation
.. ..
[]
23 1
.34. x = .3.

y
46 2
has
a) no solution, .b) unique solution, c) in.nitely many solution.

(6)
If a linear system of m equations in n variables has a unique solution, then a) m>n, b) m<n, c) m = n, .d) m . n, e) m . n, f) none of the above is true.

(7)
If a linear system of m equations in n variables has no solution, then a) m>n, b) m<n, c) m = n, d) m . n, e) m . n, .f) none of the above is true.

(8)
If a linear system of m equations in n variables has in.nitely many solutions, then a) m>n, b) m<n, c) m = n, d) m . n, e) m . n, .f) none of the above is true.

(9)
If A is an m n-matrix, then

.a) the number of pivotal columns of A is at most m,
.b) the number of pivotal columns of A is at most n,
c) the number of pivotal columns of A is at least 1,

.d) the number of pivotal columns of A could be 0.

(10)
If v1, ..., vp are linearly independent vectors in Rm, then a) p>m, b) p<m, c) p = m, d) p . m, .e) p . m, f) none of the above is true.

(11)
If v1, ..., vp are vectors in Rm and they span Rm, then a) p>m, b) p<m, c) p = m, .d) p . m, e) p . m, f) none of the above is true.

(12)
Let T : Rn . Rm be a linear map. Suppose that T is neither one-to-one nor onto, then a) m>n, b) m<n, c) m = n, d) m . n, e) m . n, .f) none of the above is true.

(13)
The following statements are true.

.a) The composition of invertible linear maps is invertible.
.b) The composition of one-to-one linear maps is one-to-one.
.c) The composition of onto linear maps is onto.
d) The composition of non-invertible linear maps is non-invertible.


(14)
Let T : Rn . Rm be a linear map. Suppose that T is onto, then a) m>n, b) m<n, c) m = n, d) m . n, .e) m . n, f) none of the above is true.

(15)
Let T : Rn . Rm be a linear map. Suppose that T is one-to-one, then a) m>n, b) m<n, c) m = n, .d) m . n, e) m . n, f) none of the above is true.

(16)
Let T : Rn . Rm be a linear map. Then

.a) The standard matrix of T is an m n matrix.
b) The standard matrix of T is an n m matrix.

.c) The range of T is a subset of Rm .
d) The range of T i