Find sequences of bases

Question

An oligoribonucleotide is treated with T1 and RP enzymes. the fragments obtained from the action of each enzyme are: a) with RP

AG^mGGAAU,AAAGU,AAG,GAT,IGC,C,C,C,C,U,U

b)with T1

ATUCCAG^m,CAAAG,CCAAG,AAUI,UUG,G,G,G

——-find the sequence of bases in this m-RNA fragment

Answer 1

A basis for a vector space is a set of vectors that is linearly independent and spans .

1) Basis for the span of given vectors

Goal: From vectors , pick a maximal independent subset.

Method (RREF):

1. Form a matrix with the vectors as columns.

2. Row-reduce to RREF.

3. The pivot columns in the RREF tell you which original columns form a basis for .

Tiny example:

.

Notice , so a basis for the span is .

2) Basis of column space / row space / null space of a matrix

Given :

Column space basis: columns of corresponding to pivot columns (found via RREF of ).

Row space basis: pivot rows of the RREF (or take the nonzero rows of RREF).

Null space basis: solve ; express free variables as parameters to get special solutions. Those special solution vectors form a basis of .

3) Orthonormal basis from a spanning set (Gram–Schmidt)

Given independent :

For : , then .

The form an orthonormal basis for .

“Basis of a sequence space”

If you literally mean sequences (like , , or the space of eventually-zero sequences):

The standard (canonical) basis is .

Every sequence can be written as (convergence in the space’s norm as appropriate).

Quick checklist (to “find a basis” fast)

Put vectors as columns → RREF → pick original pivot columns.

Need orthonormal? Run Gram–Schmidt on the indep .