M340L Final Exam, May 7 2003

1. Let tex2html_wrap_inline37 and let tex2html_wrap_inline39 . The augmented matrix [A b] is row-equivalent to tex2html_wrap_inline43 .

a) Find all solutions to Ax=b. Express your answers in parametric form.

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

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

d) Find a basis for tex2html_wrap_inline51 .

2. For each of these matrices, (i) find the determinant, (ii) state whether the matrix is invertible, and (iii) either find the inverse of the matrix (if it is invertible) or find a nonzero solution to Ax=0 (if it isn't).

a) tex2html_wrap_inline55

b) tex2html_wrap_inline57 .

3. In the space tex2html_wrap_inline59 of quadratic polynomials, let tex2html_wrap_inline61 be the standard basis, and let tex2html_wrap_inline63 be an alternate basis

a) Find the change-of-basis matrix tex2html_wrap_inline65 that converts from coordinates in the B basis to coordinates in the E basis.

b) Find the change-of-basis matrix tex2html_wrap_inline71 that converts from coordinates in the E basis to coordinates in the B basis.

c) Compute the coordinates, in the B basis, of the following four vectors: tex2html_wrap_inline79

4. Let tex2html_wrap_inline81 be defined by tex2html_wrap_inline83 , where p' is the derivative of p with respect to t. Find the matrix of this linear transformation relative to the standard basis.

5. a) Find a tex2html_wrap_inline91 matrix A whose eigenvalues are 1, 0 and -1, and whose corresponding eigenvectors are tex2html_wrap_inline97 , tex2html_wrap_inline99 and tex2html_wrap_inline101 .

b) Compute tex2html_wrap_inline103 .

6. In this problem, we model the spread of an epidemic. Let S(k) be the number of sick people in week k, and let I(k) be the number of people who are infected, but not yet sick. Each week, a sick person will infect 6 others, while an infected person will become sick. (In our model, nobody ever recovers or dies). That is,


Letting tex2html_wrap_inline111 , this boils down to tex2html_wrap_inline113 .

a) Find the eigenvalues and corresponding eigenvectors of the matrix.

b) In the long run, what will be the ratio of sick to infected (but not yet sick) people?

c) If there are 3 sick people and 1 infected person in week zero, how many sick and infected people will there be in week k?

7. Let tex2html_wrap_inline117 , tex2html_wrap_inline119 , and tex2html_wrap_inline121 . Let tex2html_wrap_inline123 .

a) Compute tex2html_wrap_inline125 .

b) Find the distance from b to the plane V.

c) Find a least-squares solution to tex2html_wrap_inline131 .

8. a) Find an orthogonal basis for the column space of tex2html_wrap_inline133 .

b) Find the projection of tex2html_wrap_inline135 onto this space.

9. True-False. 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 equation Ax=b has a solution if, and only if, b is in the span of the columns of A.

b) The equation Ax=b has a solution if, and only if, the augmented matrix [A b] has a pivot position in each row.

c) If the tex2html_wrap_inline147 matrix A has a pivot in each column, then the columns of A are linearly independent.

d) If the tex2html_wrap_inline147 matrix A has a pivot in each column, then the columns of A spen tex2html_wrap_inline159 .

e) Every linear transformation from tex2html_wrap_inline161 to tex2html_wrap_inline159 can be represented by an tex2html_wrap_inline147 matrix.

f) If A, B and C are matrices such that the product ABC makes sense, then tex2html_wrap_inline175 .

g) If the determinant of a square matrix A is zero, then A is invertible.

h) Given vectors tex2html_wrap_inline181 , the set of all linear combinations of these vectors is a subspace of tex2html_wrap_inline161 .

i) If two matrices are row-equivalent, then their column spaces have the same dimension.

j) The row space of a matrix has the same dimension as the column space.

k) tex2html_wrap_inline185 is a subspace of tex2html_wrap_inline187 .

l) The range of T(x) = Ax is the same as the column space of A.

m) If tex2html_wrap_inline193 , then tex2html_wrap_inline195 is a basis for H.

n) The dimension of the null space of a matrix is the number of free variables in the equation Ax=0.

o) If A is a tex2html_wrap_inline203 matrix, then the null space of A is at least 2-dimensional.

p) A number c is an eigenvalue of A if and only if det(A-cI)=0.

q) If the characteristic polynomial of a tex2html_wrap_inline213 matrix A is tex2html_wrap_inline217 , then A is diagonalizable.

r) Every tex2html_wrap_inline221 matrix has at least one eigenvalue, but it may be complex.

s) If tex2html_wrap_inline223 , with D diagonal, then each column of P is an eigenvector of A.

t) If Ax=0, then x is in tex2html_wrap_inline235 .

u) If two nonzero vectors are orthogonal, they are linearly independent.

v) If two nonzero vectors are linearly independent, they are orthogonal.

w) If tex2html_wrap_inline237 , where tex2html_wrap_inline239 and tex2html_wrap_inline241 , then tex2html_wrap_inline243 .

x) The equation Ax=b always has a least-squares solution, no matter what A and b are.

y) If tex2html_wrap_inline251 , then Ax=b has a unique least-squares solution.

Lorenzo Sadun
Wed May 7 13:25:37 CDT 2003