M340L Second Midterm Exam, March 7, 2003

1. Consider the tex2html_wrap_inline37 matrix


a) What is the rank of A? [Note: the row reduction of A can be done using integers only. If you start getting fractions, you probably made a mistake]

b) Find a basis for the column space of A.

c) Find a basis for the null space of A.

2. Consider the linear transformation tex2html_wrap_inline49 defined by
tex2html_wrap_inline51 .

a) Find the matrix of this linear transformation.

b) Is T one-to-one? Is T onto?

c) Let S be a rectangle whose area is 2. What is the area of T(S)?

3. Compute the following determinants:

a) tex2html_wrap_inline61

b) tex2html_wrap_inline63

c) tex2html_wrap_inline65

4. Consider the matrix tex2html_wrap_inline67 , that factorizes as A = LU, where tex2html_wrap_inline71 and tex2html_wrap_inline73 . Let tex2html_wrap_inline75 . Solve the problem tex2html_wrap_inline77 in two steps (Note: you do NOT get any credit for solving the full equations directly):

a) Solve tex2html_wrap_inline79 for tex2html_wrap_inline81 , and then

b) Solve tex2html_wrap_inline83 for tex2html_wrap_inline85 . Does this solution also satisfy tex2html_wrap_inline77 ?

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 rank of a matrix is the dimension of its null space.

b) Swapping two rows of a matrix does not change the determinant of that matrix.

c) If A and B are square matrices of the same size, then tex2html_wrap_inline93 .

d) If an tex2html_wrap_inline95 matrix has an inverse, then the columns of that matrix span tex2html_wrap_inline97 .

e) If the determinant of a (square) matrix is zero, then its columns are linearly dependent.

f) The span of 3 vectors in tex2html_wrap_inline97 is a 3-dimensional subspace of tex2html_wrap_inline97 .

g) Let A be an tex2html_wrap_inline105 matrix. The dimension of Col(A) plus the dimension of Null(A) equals 5.

h) If two vectors tex2html_wrap_inline111 and tex2html_wrap_inline113 lie in a subspace H of tex2html_wrap_inline97 , then every linear combination of tex2html_wrap_inline111 and tex2html_wrap_inline113 also lies in H.

i) If a tex2html_wrap_inline125 matrix has rank 3, then its null space is 1-dimensional.

j) A basis for a subspace H is a linearly independent set of vectors whose span is H

Lorenzo Sadun
Fri Mar 7 15:11:05 CST 2003