MTH 007 Linear Algebra
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Linear Algebra
MTH007: Ch 1.3 Vector Equations
1. Let A = 
10一2 
一一32(4) ．(、) and b = 
41一4．(、) . Denote the columns of A by a1 , a2 , a3 , and let W = span(a1 , a2 , a3 ).
(a) Is b in (a1 , a2 , a3 ) ? How many vectors are in (a1 , a2 , a3 )?
(b) Is b in W ? How many vectors are in W .
(c) Show that a1 is in W .
2. Construct a 3 ( 3 matrix A, with nonzero entries and a vector b è R3 such that b is not in the set spanned by the columns of A.
3. Let v1 ,．．．, vk be points in R3 and suppose that for j = 1,．．．, k, an object with mass mj is located at point vj . Physicists call such objects point masses. The total mass of the system point masses is
m = m1 +．．．+ mk .
The center of gravity (or center of mass) of the system is
= 
(m1v1 +．．．+ mkvk).
Compute the center of the gravity of the system consisting of the following point masses. Determine if 
is in Span(v1 ,．．．, vk) ?
|
points |
mass |
|
v1 = (5, 一4, 3) v2 = (4, 3, 一2) v3 = (一4, 一3, 一1) v4 = (一9, 8, 6) |
2g 5g 2g 1g |
MTH007: Ch 1.4 The Matrix Equation Ax = b
1. Let A = / 2一6 一31 、 and b = /b(b)2(1) 、.
❼ Show that the equation Ax = b does not have a solution for all possible b ❼ Describe the set of all b for which Ax = b does have a solution.
╱ 0(1) 、 ╱ 0一1 、 ╱ 0(1) 、
2. Let v1 = ．．( 一01 ．．．, v2 = ．．( 1(0) ．．．, v3 = ．．( 0一1 ．．．. Explain whether (v1 , v2 , v3 ) span R4 ?
3. Let A be a 3 ( 4 matrix. Let y1 and y2 be vectors in R3 and w = y1 + y2 . Suppose y1 = Ax1, y2 = Ax2
for some vectors x1 and x2 in R4 . Show that Ax = w is consistent.
2022-09-22