Math 20F
Test 1
100 points
January 28, 2009
--------------------------------------------------------------------------------------------------Notation: All scalars and matrix entries are in the set R of real numbers.
Points: (1),(2) 12 pts each (3A),(3B),(3C),(4) 15 pts each (3D) 10 pts (3E) 6 pts
Directions: Show all work and justify all answers, except in problems (1) and (2).
----------------------------------------------------------------------------------------------------(1) Let S = { v1 , . . . , vn } be a set of n vectors in Rm . Complete the following
sentences in such a way as to provide precise, explicit definitions:
(A) The set S is called linearly dependent if ___________________________.
(B) The span of S is defined to be the set ______________________________.
-----------------------------------------------------------------------------------------------(2) Let T : Rn  Rm be a function (i.e., a transformation). Complete the following
sentences in such a way as to provide precise, explicit definitions:
(A) T is called onto if _________________________________________________.
(B) T is called one-to-one if ___________________________________________.
-------------------------------------------------------------------------------------------------(3) Define A
0 0 0 1
A= 1 0 1 0
2 1 5 1
and U and b as follows:
2
1 0 1 0 2
2
,
U= 0 1 3 0 1
7
0 0 0 1 2
,
-3
b = 1
5
.
(Note that the matrices A and U are 3 by 5, while the vector b is 3 by 1.)
(A) Show (step by step) that U is the reduced row echelon form of A.
(B) Find two 5 by 1 vectors c and d such that Span{c, d} is the set of
all solutions x in R5 to the equation Ax = 0 .
(C) Find the set of all solutions x in R5 to the equation Ax = b .
Hint: Part (B) may be of use.
(D) Is every 3 by 1 vector equal to some linear combination
of the five columns of A ? Explain your answer.
(E) What is the number in the lower right position of the 5 by 5 matrix (AT U)T ?
--------------------------------------------------------------------------------------------------(4) Let T : R3  R5 be a one- to-one linear transformation and let L denote a
straight line in R3 . Show that the image T(L) in R5 is also a line.