Consider the following subsets of R^3 I. A = {(1,2,3), (1,-1,0)} II. B = {(1,1,0
ID: 3032763 • Letter: C
Question
Consider the following subsets of R^3 I. A = {(1,2,3), (1,-1,0)} II. B = {(1,1,0), (1,2,3), (0,1,0)} III. C = {(1,1,0), (1,2,3), (0,1,-1)} IV. D = {(1,1,0), (1,2,3), (0,0,1)} Find all sets of vectors which is linearly independent. Find all sets of vectors which forms a basis for R^3 Suppose [v]_D = [1 -2 3]. Find [v]_s where S = ((1,0,0), (0,1,0), (0,0,1)} Find [w]_D if w = [-1 -4 -4] Find the dimension of the subspace of R^4 spanned by the vectors {(1,1,0,0), (1,2, -2,1), (3,6, -5,4), (1,0,2,3)}Explanation / Answer
#(5)
The set S = {v1, v2, v3, v4} of vectors in R4 is linearly independent if the only solution of
c1v1 + c2v2 + c3v3 + c4v4 = 0
I'm going to write them horizontally
c1(1,1,0,0) +c2(1,2,-2,1) +c3 (3,6,-5,4) +c4 ( 1, 0, 2, 3 )
Solve the resulting homogeneous system of equations using any method you know
c1+ c2 + 3c3 + c4=0 ........................(1)
c1 + 2c2 + 6c3 = 0 ............................(2)
-2c2 - 5c3 + 2c4 = 0 ...........................(3)
c2 + 4c3 + 3c4 = 0 ...............................(4)
You get four basic variables and one free which is c4 set it equal to t for example
c1 = 0
c2 = 0
c3 = 0
c4 = 0
set T = { v1, v2, v3,v4}
Since the set T = {v1, v2, v3,v4} is linearly independent and it spans span S, then the set
T={(1,1,0,0), (1,2,-2,1), (3,6,-5,4),(1,0,2,3) } forms a basis for span S which is of dimension 4.