M340L Final Exam Solution, May 7 2003

1. Let tex2html_wrap_inline117 and let tex2html_wrap_inline119 . The augmented matrix [A b] is row-equivalent to tex2html_wrap_inline123 .

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

The row-reduced equations read tex2html_wrap_inline127 , tex2html_wrap_inline129 , tex2html_wrap_inline131 , while tex2html_wrap_inline133 and tex2html_wrap_inline135 are free ( tex2html_wrap_inline137 and tex2html_wrap_inline139 ). Thus


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

The first, 2nd and 5th columns are pivot columns, so our basis is


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

This is the same as part (a), only with zero on the right hand side. Our basis is tex2html_wrap_inline149 .

d) Find a basis for tex2html_wrap_inline151 .

This is the same thing as the null space of A, so the answer is the same as (c).

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_inline157

The determinant is 0, the matrix is NOT invertible, and a nontrivial solution to Ax=0 is tex2html_wrap_inline161 .

b) tex2html_wrap_inline163 .

The determinant is -1 (expand about the first column, and then about the last column), the matrix is invertible, and the inverse (obtained by row-reducting [A I]) is tex2html_wrap_inline167 .

3. In the space tex2html_wrap_inline169 of quadratic polynomials, let tex2html_wrap_inline171 be the standard basis, and let tex2html_wrap_inline173 be an alternate basis

a) Find the change-of-basis matrix tex2html_wrap_inline175 that converts from coordinates in the B basis to coordinates in the E basis.


b) Find the change-of-basis matrix tex2html_wrap_inline183 that converts from coordinates in the E basis to coordinates in the B basis.

tex2html_wrap_inline189 (computed by row-reducing tex2html_wrap_inline191 .)

c) Compute the coordinates, in the B basis, of the following four vectors: tex2html_wrap_inline195

Just multiply tex2html_wrap_inline183 by the coordinates of these vectors in the E basis to get: tex2html_wrap_inline201

4. Let tex2html_wrap_inline203 be defined by tex2html_wrap_inline205 , where p' is the derivative of p with respect to t. Find the matrix of this linear transformation relative to the standard basis.

Since tex2html_wrap_inline213 , tex2html_wrap_inline215 and tex2html_wrap_inline217 , the matrix is tex2html_wrap_inline219 .

5. a) Find a tex2html_wrap_inline221 matrix A whose eigenvalues are 1, 0 and -1, and whose corresponding eigenvectors are tex2html_wrap_inline227 , tex2html_wrap_inline229 and tex2html_wrap_inline231 .

tex2html_wrap_inline233 where tex2html_wrap_inline235 and tex2html_wrap_inline237 . Computing tex2html_wrap_inline239 and multiplying gives tex2html_wrap_inline241 .

b) Compute tex2html_wrap_inline243 .

tex2html_wrap_inline245 . But tex2html_wrap_inline247 is easily seen to equal D, so tex2html_wrap_inline251 .

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_inline259 , this boils down to tex2html_wrap_inline261 .

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

Eigenvalues are tex2html_wrap_inline263 and tex2html_wrap_inline265 and eigenvectors are tex2html_wrap_inline267 and tex2html_wrap_inline269 (or any nonzero multiple of these choices).

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

In the long run, the coefficient of tex2html_wrap_inline271 dominates, so the ratio of sick to infected approaches that of tex2html_wrap_inline271 , namely 1:2.

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?

tex2html_wrap_inline277 , so tex2html_wrap_inline279 .

7. Let tex2html_wrap_inline281 , tex2html_wrap_inline283 , and tex2html_wrap_inline285 . Let tex2html_wrap_inline287 .

a) Compute tex2html_wrap_inline289 .

Since tex2html_wrap_inline291 and tex2html_wrap_inline293 are orthogonal, tex2html_wrap_inline295 .

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

The distance is tex2html_wrap_inline301 .

c) Find a least-squares solution to tex2html_wrap_inline303 .

We have already seen that the projection of b is tex2html_wrap_inline307 , so our least-squares solution is tex2html_wrap_inline309 . You can also get this answer by solving tex2html_wrap_inline311 .

8. a) Find an orthogonal basis for the column space of tex2html_wrap_inline313 .




b) Find the projection of tex2html_wrap_inline321 onto this space.

The vectors tex2html_wrap_inline323 give an orthogonal basis for this column space, so tex2html_wrap_inline325 , and tex2html_wrap_inline327 . The vector b was already IN the column space, so its projection is itself.

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.

False. It has a solution if there is NOT a pivot in the last COLUMN.

c) If the tex2html_wrap_inline341 matrix A has a pivot in each column, then the columns of A are linearly independent.


d) If the tex2html_wrap_inline341 matrix A has a pivot in each column, then the columns of A spen tex2html_wrap_inline353 .

False. Having a pivot in each column implies the columns are linearly independent. Having a pivot in each ROW implies that they span tex2html_wrap_inline353 .

e) Every linear transformation from tex2html_wrap_inline357 to tex2html_wrap_inline353 can be represented by an tex2html_wrap_inline341 matrix.


f) If A, B and C are matrices such that the product ABC makes sense, then tex2html_wrap_inline371 .


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

False. If the determinant is zero, the matrix is NOT invertible.

h) Given vectors tex2html_wrap_inline377 , the set of all linear combinations of these vectors is a subspace of tex2html_wrap_inline357 .

True. The span of several vectors is always a subspace.

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.

True. Both dimensions equal the rank of the matrix.

k) tex2html_wrap_inline381 is a subspace of tex2html_wrap_inline383 .

False. tex2html_wrap_inline381 does not sit inside tex2html_wrap_inline383 .

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


m) If tex2html_wrap_inline393 , then tex2html_wrap_inline395 is a basis for H.

False. They may not be linearly independent.

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_inline403 matrix, then the null space of A is at least 2-dimensional.

False. This would be true for a tex2html_wrap_inline407 matrix, not a tex2html_wrap_inline403 .

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_inline417 matrix A is tex2html_wrap_inline421 , then A is diagonalizable.

True. The eigenvalues are all distinct.

r) Every tex2html_wrap_inline425 matrix has at least one eigenvalue, but it may be complex.


s) If tex2html_wrap_inline233 , with D diagonal, then each column of P is an eigenvector of A.


t) If Ax=0, then x is in tex2html_wrap_inline439 .

False. x is in tex2html_wrap_inline151 .

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


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

False. (e.g., tex2html_wrap_inline445 and tex2html_wrap_inline447 .)

w) If tex2html_wrap_inline449 , where tex2html_wrap_inline451 and tex2html_wrap_inline453 , then tex2html_wrap_inline455 .


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

True. Least-squares solutions always exist.

y) If tex2html_wrap_inline463 , then Ax=b has a unique least-squares solution.

False. Uniqueness has to do with the columns of A being linearly independent.

Lorenzo Sadun
Wed May 7 13:26:29 CDT 2003