9.3. EXERCISES 203

50. Using Problem 48, find the general solution to the following linear system.

1 2 3 2 10 2 1 1 21 4 4 3 30 2 1 1 2



x1

x2

x3

x4

x5

=

67

137



51. Suppose Ax = b has a solution. Explain why the solution is unique precisely whenAx = 0 has only the trivial (zero) solution.

52. Show that if A is an m×n matrix, then ker(A) is a subspace.

53. Verify the linear transformation determined by the matrix of 9.2 maps R3 onto R2

but the linear transformation determined by this matrix is not one to one.

54. You are given a linear transformation T : Rn→ Rm and you know that

T ai = bi

where(

a1 · · · an

)−1exists. Show that the matrix A of T with respect to the

usual basis vectors (T x = Ax) must be of the form(b1 · · · bn

)(a1 · · · an

)−1

55. You have a linear transformation T and

T

 12−6

=

 513

 ,T

 −1−15

=

 115

 , T

 0−12

=

 53−2

Find the matrix of T . That is find A such that T x = Ax.

56. You have a linear transformation T and

T

 11−8

=

 131

 ,T

 −106

=

 241

 ,T

 0−13

=

 61−1

Find the matrix of T . That is find A such that T x = Ax.

57. You have a linear transformation T and

T

 13−7

=

 −313

 ,T

 −1−26

=

 13−3

 , T

 0−12

=

 53−3

Find the matrix of T . That is find A such that T x = Ax.