M340L 3rd Midterm With Solutions, April 7, 2003

1. The following two tex2html_wrap_inline51 matrices are row-equivalent:


a) What is the rank of A?

There are 3 pivots, so the rank is 3.

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

The pivots are in the 1st, 2nd and 4th columns, so a basis is
tex2html_wrap_inline59 . Note that the 1st, 2nd and 4th columns of B are NOT a basis for Col(A). They are a basis for Col(B).

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

After row-reduction our equations become: tex2html_wrap_inline69 , tex2html_wrap_inline71 , tex2html_wrap_inline73 , and tex2html_wrap_inline75 , so all solutions are multiples of the single basis vector tex2html_wrap_inline77 .

d) Find a basis for the row space of A.

Since Row(A)=Row(B), our basis is tex2html_wrap_inline83 . Note that the first three rows of A are NOT a basis, as the third is the sum of the first two.

2. Let V=Span tex2html_wrap_inline89 .

a) What is the dimension of V?

V is the column space of tex2html_wrap_inline95 , which row-reduces to tex2html_wrap_inline97 . There are 3 pivots, so the dimension is 3.

b) Find a basis for V.

The pivots are in the first 3 columns, so the first three vectors will do, namely tex2html_wrap_inline101

c) Let W be the subspace of tex2html_wrap_inline105 spanned by the polynomials tex2html_wrap_inline107 , tex2html_wrap_inline109 , tex2html_wrap_inline111 and tex2html_wrap_inline113 . What is the dimension of W? Find a basis for W.

Once you write down the coordinates, this is the EXACT SAME PROBLEM. The dimension is 3, and the basis is the set of vectors whose coordinates are the answer to part (b), namely tex2html_wrap_inline119 .

3. Consider the following basis for tex2html_wrap_inline121 :


a) Find the change-of-basis matrix tex2html_wrap_inline125 that converts from coordinates in the B basis to coordinates in the standard (E) basis.

The columns of tex2html_wrap_inline125 are just the vectors themselves:


b) Find the change-of-basis matrix tex2html_wrap_inline135 that converts from coordinates in the standard basis to coordinates in the B basis.

tex2html_wrap_inline139 , as computed by row-reducing tex2html_wrap_inline141 .

c) Let tex2html_wrap_inline143 . Compute tex2html_wrap_inline145 .

tex2html_wrap_inline147 . You can check that x is indeed equal to tex2html_wrap_inline151 .

d) Consider the basis tex2html_wrap_inline153 , tex2html_wrap_inline155 , tex2html_wrap_inline157 for the vector space tex2html_wrap_inline159 . Find the coordinates of the polynomial tex2html_wrap_inline161 in this basis.

Once you write this in coordinates (with respect to the standard basis), this is the same problem as (c), and has the same answer: tex2html_wrap_inline163

4. a) Are the polynomials tex2html_wrap_inline165 , tex2html_wrap_inline167 and tex2html_wrap_inline169 linearly independent?

Use coordinates to convert this into a problem in tex2html_wrap_inline121 . Row-reducing tex2html_wrap_inline173 gives tex2html_wrap_inline175 , which only has 2 pivots. Since there is not a pivot in the last column, the three vectors are NOT linearly independent.

b) Do the polynomials tex2html_wrap_inline177 , tex2html_wrap_inline179 and tex2html_wrap_inline181 span tex2html_wrap_inline159 ?

Likewise, use coordinates. The matrix tex2html_wrap_inline185 DOES row-reduce to something with a pivot in each row, so the vectors DO span the space. (Instead of row-reducing, you could also compute the determinant).

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 row space.

TRUE. It's also the dimension of the column space.

b) Let tex2html_wrap_inline187 be vectors in a vector space V. For Span tex2html_wrap_inline191 to be a subspace of V, the vectors must be linearly independent.

FALSE. If the vectors are linearly dependent, then the dimension of Span tex2html_wrap_inline191 will be less than 3, but it will still be a subspace.

c) If A and B are row-equivalent matrices, then Col(A)=Col(B).

FALSE. See Problem 1 for a counterexample.

d) If A and B are row-equivalent matrices, then Null(A)=Null(B).

TRUE. The equations Ax=0 and Bx=0 are equivalent.

e) If A and B are row-equivalent matrices, then Row(A)=Row(B).

TRUE. The rows of each can be obtained from the rows of the other.

f) Let tex2html_wrap_inline219 be a collection of four vectors in tex2html_wrap_inline221 . One of the tex2html_wrap_inline223 's can be written as a linear combination of the others.

TRUE. tex2html_wrap_inline221 has dimension 3, so any collection of 4 or more vectors is linearly dependent, so one of the vectors is a linear combination of the others.

g) If a collection of five polynomials spans tex2html_wrap_inline227 , then it forms a basis for tex2html_wrap_inline227 .

FALSE. tex2html_wrap_inline227 is only 4-dimensional, so any collection of 5 vectors will fail to be linearly independent.

h) Every change-of-basis matrix is invertible.

TRUE.  The inverse of P_{BC} is P_{CB}.

i) If a linearly dependent set of vectors spans tex2html_wrap_inline121 , then there must be at least 4 vectors in the set.

TRUE. A linearly dependent set of 3 (or fewer) vectors can't span tex2html_wrap_inline121 .

j) If tex2html_wrap_inline237 is a linearly independent set of vectors in V, and tex2html_wrap_inline241 is a spanning set for V, then n>k.

FALSE. All you can say is that tex2html_wrap_inline247 .

Lorenzo Sadun
Mon Apr 7 11:40:02 CDT 2003