Name________________________
Student I.D.___________________
Math 2250-1
Quiz 6 SOLUTIONS
October 5, 2012
1a) What is the span of a collection of vectors v1, v2, ... vn ?
the collection of all possible linear combinations....in math notation
span v1, v2, ... vn = c1v1 C c2 v2 C ...C cn vn s.t. c1, c2,...cn 2 =
(1 point)
1b) What does it mean for vectors v1, v2, ... vn to be linearly independent?
(1 point)
the only linear combination of v1, v2, ... vn which equals zero is the one for which all the linear combination coefficients
are zero.....in math notation
c1v1 C c2 v2 C ...C cn vn = 0 0 c1 = c2 =...= cn = 0.
1c) What is a basis for a vector space/subspace W ?
(1 point)
a linearly independent collection of vectors that spans W.
2a) Find a basis for the solution space to homogeneous matrix equation A x = 0 , where A is the matrix on the left below,
and its reduced row echelon form is on the right:
1 K2 0 K2
1
2 K4 1 K3
0
1 K2 2
0 K4
2 K4 0 K4
1 K2 0 K2 0
/
2
0
0 1
1 0
0
0 0
0 1
0
0 0
0 0
.
(6 points)
The augmented matrices just have a column of zeroes appended on the right, so we can backsolve the reduced row echelon
form to get x5 = 0, x4 = t, x3 =Kt, x2 = s, x1 = 2 s C 2 t. In linear combination form this becomes
x1
x2
x3
x4
x5
2 sC2 t
2
s
0
K
t
K
=
=t 1 Cs
t
1
0
0
Thus a basis consists of the two vectors 2, 0,K1, 1, 0 T, 2, 1, 0, 0, 0
2b) What is the dimension of the solution space in 2a?
Since there are two vectors in a basis the dimension is the number 2.
T
2
1
0
0
0
.
.
(1 point)