UNC STOR-BIOS Boot Camp 2015 - Algebra
HW 4 Solutions
Wanyi Chen
August 10, 2015
Problem 3.5
For any x N (A), we have Ax = 0. Then A0 Ax = 0. Hence x N (A0 A) as well, which
indicates that N (A) N (A0 A).
On the other hand, for any x N (A0 A), A0 Ax = 0.

UNC STOR-BIOS Boot Camp 2015 - Algebra
HW 5 Solutions
Wanyi Chen
August 10, 2015
Problem 4.2
By the definition of best least square fit, we want to find C such that E 2 (C) = (y1 C)2 +
. + (ym C)2 is minimized, and hence the result follows.
Problem 5.1
(a

UNC STOR-BIOS Boot Camp 2015 - Algebra
HW 1 Solutions
Wanyi Chen
August 4, 2015
Problem 2.1
Check the definition of vector spaces.
Problem 2.2
(i) Let A = cfw_(x, x) : x Z. Then for any a, b Z, (a, a) (b, b) = (a b, a b) A.
Yet 12 (1, 1) = ( 21 , 12 )
/

UNC STOR-BIOS Boot Camp 2015 - Algebra
HW 3 Solutions
Wanyi Chen
August 10, 2015
Problem 2.6
(a)
R1
1
xdx = 0
(b) Notice that h1, xi = 0 and hx, x2 i = 0 and obviously x2 cannot be written as a linear
combination of 1 and x. Therefore cfw_1, x, x2 are li

UNC STOR-BIOS Boot Camp 2015 - Algebra
HW 2 Solutions
Wanyi Chen
August 5, 2015
Problem 2.5
Check the definition of inner product.
Problem 2.6
(a)
R1
1
xdx = 0
(b) Notice that h1, xi = 0 and hx, x2 i = 0 and obviously x2 cannot be written as a linear
comb

UNC STOR-BIOS Boot Camp 2015 - Algebra
HW 6 Solutions
Wanyi Chen
August 11, 2015
Problem 6.1
Proof by Induction.
(i) Obviously the claim holds for 1 1 triangular matrices.
(ii) Assume that it holds for kk triangular matrices. Let A be a (k+1)(k+1) triangu