April 3, 2006
Lecturer Dmitri Zaitsev
Trinity Term 2006
Course 2E1 2005-06 (SF Engineers & MSISS & MEMS)
S h e e t 17
Due: in the tutorial sessions next Wednesday/Thursday
Exercise 1
Find the coordinates of the vector v with respect to the basis v1 , . . . , vn (i.e. the coefficients c1 , . . . , cn in the representation v = c1 v1 + · · · + cn vn :
(i) v = (2, 5), v1 = (2, −1), v2 = (1, 2);
(ii) v = (2, −1, 3), v1 = (1, −1, 0), v2 = (1, 1, 0), v3 = (1, 1, 2);
(iii) v = (1, 2, 1, 2), v1 = (1, 0, 0, 0), v2 = (1, 1, 0, 0), v3 = (0, 0, 6, 0), v4 = (−1, 1, 1, 1).
Exercise 2
Find bases and dimensions for the row, column and null spaces of the matrix:
(i) ( 2 −1 );
2
(ii)
;
−1
1 −3
(iii)
;
−1 3
1 −3 0
(iv)
;
−1 3 1


1 −3
(v)  −1 3 ;
2
0