M340L 3rd Midterm, April 7, 2003

1. The following two tex2html_wrap_inline26 matrices are row-equivalent:


a) What is the rank of A?

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 the row space of A.

2. Let V=Span tex2html_wrap_inline40 .

a) What is the dimension of V?

b) Find a basis for V.

c) Let W be the subspace of tex2html_wrap_inline48 spanned by the polynomials tex2html_wrap_inline50 , tex2html_wrap_inline52 , tex2html_wrap_inline54 and tex2html_wrap_inline56 . What is the dimension of W? Find a basis for W.

3. Consider the following basis for tex2html_wrap_inline62 :


a) Find the change-of-basis matrix tex2html_wrap_inline66 that converts from coordinates in the B basis to coordinates in the standard (E) basis.

b) Fina the change-of-basis matrix tex2html_wrap_inline72 that converts from coordinates in the standard basis to coordinates in the B basis.

c) Let tex2html_wrap_inline76 . Compute tex2html_wrap_inline78 .

d) Consider the basis tex2html_wrap_inline80 , tex2html_wrap_inline82 , tex2html_wrap_inline84 for the vector space tex2html_wrap_inline86 . Find the coordinates of the polynomial tex2html_wrap_inline88 in this basis.

4. a) Are the polynomials tex2html_wrap_inline90 , tex2html_wrap_inline92 and tex2html_wrap_inline94 linearly independent?

b) Do the polynomials tex2html_wrap_inline96 , tex2html_wrap_inline98 and tex2html_wrap_inline100 span tex2html_wrap_inline86 ?

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.

b) Let tex2html_wrap_inline104 be vectors in a vector space V. For Span tex2html_wrap_inline108 to be a subspace of V, the vectors must be linearly independent.

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

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

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

f) Let tex2html_wrap_inline130 be a collection of four vectors in tex2html_wrap_inline132 . One of the tex2html_wrap_inline134 's can be written as a linear combination of the others.

g) If a collection of five polynomials spans tex2html_wrap_inline136 , then it forms a basis for tex2html_wrap_inline136 .

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

i) If a linearly dependent set of vectors spans tex2html_wrap_inline62 , then there must be at least 4 vectors in the set.

j) If tex2html_wrap_inline142 is a linearly independent set of vectors in V, and tex2html_wrap_inline146 is a spanning set for V, then n>k.

Lorenzo Sadun
Mon Apr 7 11:37:59 CDT 2003