# Practice Exam #1 ```Practice Exam #1
Math 2270, Spring 2005
Problem 1. Consider the following matrix equation Ax = b:

   
2 2
1
x
2
 0 1 −1   y  =  3  .
1 1
1
z
0
(a) Find the solution to this system using elementary row operations.
(b) Find the inverse of the matrix A.
(c) For invertible matrices A, there is a unique solution to Ax = b, and a formula for this solution
which uses inverse matrix. Use this formula to resolve the system in part (a). (You should get the
(d) Explain (in general) why the formula you used in part (c) gives a correct solution, and why
this is the only solution.
Problem 2. Suppose that T : R3 → R2 is a linear map and that we know what T does to the
columns of the ,matrix A from Problem 1:
 


 
2
1
2
−1
0
1












1
−1
0
, T
, and T
.
T
=
=
=
1
1
1
1
1
1
 
2


3  . (Hint: Use the abstract definition of what a linear function is, together with
Find T
0
Problem 3. We consider a linear transformation T (x) = Ax, for the following matrix A with its
reduced row-echelon form:




1 0 0
1
1
2
1 −1
3
0

 1 −1
0 −4 
2 −1
1 

 and rref (A) =  0 1 0
A=
 0 0 1 −1 −2 
 0
3 −5
5 −2 
0 0 0
0
0
1
0
0
1
1
(a) Find a basis for the image of T .
(b)Express the fifth column of A as a linear combination of the basis vectors you found in part
(a).
(c) Find a basis for the kernel of T . Verify that your set of vectors is linearly independent.
(d) State the Rank-Nullity theorem and verify that this theorem hold for the matrix A above.
Problem 4. Prove that any linear transformation T : R2 → R2 , that is given by the projection of
a vector on a line L has the form T (x) = Ax, where
u21 u1 u2
, u21 + u22 = 1.
A=
u1 u2
u22
Problem 5. (a) Give the definition of the kernel of a matrix n &times; m.
(b) Give the definition of basis of a subspace of Rn .