## M340L First Midterm Exam, February 12, 2003

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1. Find all solutions to the system of equations:

Is the set of solutions empty, a point, a line, a plane, or some other
geometric shape? If it is a line or a plane, express this set in parametric
form.

2. Consider the matrix and vector

a) Is
in the span of the columns of *A*? If so, write
explicitly as a linear combinations of the columns.

b) Are the columns of *A* linearly independent? If not, write the
zero vector as a nontrivial linear combination of the columns of *A*.

3. Consider the linear transformation
given by

a) Find the matrix of *T*.

b) Is *T* one-to-one? Why or why not?

c) Is *T* onto? Why or why not?

4. Indicate whether each of these matrices is invertible. If it is, find
the inverse. If it isn't invertible, explain why.

a)

b)

5. Indicate whether each of these statements is true or false. If a statement
is sometimes true and sometimes false, write ``false''. You do NOT have
to justify your answers. There is no penalty for wrong answers, so go ahead
and guess if you are unsure of your answer.

a) The columns of a
matrix must be linearly dependent.

b) The columns of a
matrix must span
.

c) If two matrices *A* and *B* have different sizes, then the
sum *A* + *B* is not defined.

d) If two matrices *A* and *B* have different sizes, then the
product *AB* is not defined.

e) If *A* and *B* are inverses, then *AB*=*BA*.

f) If a set
of 2 or more vectors is linearly dependent, then one of the vectors is
a linear combination of the others.

g) If the columns of a
matrix are linearly independent, then they span
.

h) If the columns of an
matrix are linearly independent, then they span
.

i) If a
matrix *A* has exactly 3 pivots, then the linear transformation
is one-to-one.

j) If a
matrix *A* has exactly 3 pivots, then the linear transformation
is onto

Lorenzo Sadun

* Wed Feb 12 08:37:31 CST 2003*