Su SU19
M303
Practice Midterm
1. Multiple Choice.
(a) Let A be any 5 × 4 matrix, fill in the following blanks with one of the three choices:
(A) must
(B) must not
(C) may or may not
i. A system Ax = b
be consistent.
ii. If a system Ax = b is consistent, it
have a unique solution.
iii. The column vectors of A
be linearly dependent.
iv. The column vectors of A
span the entire R5 .
v. The linear transformation x 7→ Ax
be onto.
vi. The linear transformation x 7→ Ax
be one-to-one.
(b) Let A be a m × n matrix. If Ax = b has a unique solution, choose ALL the correct
statements from below.
(A)
(B)
(C)
(D)
The row echlon form of A must have a pivot in each row.
The row echlon form of A must have a pivot in each column.
The vector b must be in the span of the column vectors of A.
The vector b can be uniquely written as a linear combination of the column
vectors of A.
(c) If a n × n matrix A is not invertible, which of the following must be true? Choose
ALL the correct statements.
(A)
(B)
(C)
(D)
Ax = b is inconsistent for every b ∈ Rn .
Ax = b either has no solution or infinitely many solutions.
The column vectors of A are linearly independent.
The determinant of A is zero.
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2. Short answer.
(a) Give an example of a 3 × 3 matrix A so that the solution set of Ax = 0 is a plane
in R3 .
(b) Give an example of a linear transformation T : R3 → R2 that is onto.
  
x1

T x2  = 
x3




 
1
0



(c) Given vectors v1 = −1 and v2 = 1 . Find a vector v3 so that v1 , v2 , v3
0
−1
3
span the entire R .


1 3 2 5
2 3 6
, B = 5 2 1 3
(d) Let A =
4 1 2
0 4 1 2
×
The dimension of the matrix AB is
The (2, 3) entry of AB is
.
.
(e) Let A, B, C, M be n × n matrices, solve M from the matrix equation CBM A = I.
M=
2
   
    
 
a
1 
0
 1
 1










0 , 1
0 , 1 , v is linearly
3. If v = b is a vector such that v ∈ span
, and




c
2
2
0
1
independent. What can you say about the values of a, b and c.
4. Find the matrix of the linear transformation T : R2 → R2 that rotates the plane
counterclockwise by 3π/2.
5. Let A be a n × n invertible matrix. Prove that
det(A−1 ) =
3
1
det(A)


3 −1 1 2
0 3 0 1

6. Let A = 
5 1 0 2.
2 1 0 1
(a) Find the determinant of A. Based on your answer, is A invertible or not?
(b) Based on your answer for part (a), does Ax = 0 have any nontrivial solution?
7. Let T : R2 → R2 be a linear transformation defined as T (x1 , x2 ) = (4x1 + x2 , 2x1 + 3x2 ).
(a) If S is a region in R2 with area 2, what is the area of the image of S under T ?
(b) Is T invertible? If so, find the standard matrix for T −1 .
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