LINEAR ALGEBRA - VECTOR SPACES
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Let W be the subspace of R^4 spanned by the set S = {[1, 1, 2, -1], [1, 2, 1, 1], [1, 4, -1, 5], [1, 0, 4, -1], [2, 5, 0, 2]} Find an ordered subset B of S that forms a basis for W. For the ordered basis B you found, give the coordinate vector [w]B for each vector w in S.

W =[V1,V2,V3,V4]
V1=	V2=	V3=	V4=	V5=
1	1	1	1	2
1	2	4	0	5
2	1	-1	4	0
-1	1	5	-1	2
W IS IN R4 …SO…
WE NEED 4 AND ONLY 4 L.I. VECTORS TO FORM A BASIS FOR IT
IF IT SPANS R4 ..IF IT DOES NOT SPAN R4 , BUT SPANS ONLY R3 OR LOWER
THEN WE NEED ONLY THOSE NUMBER OF L.I. VECTORS TO FORM A BASIS ..
THE STD METHOD IS TO REDUCE THE ABOVE IN RREF & CHECK
NR2=R2-R1….NR3=R3-2R1….NR4=R4+R1
1	1	1	1	2
0	1	3	-1	3
0	-1	-3	2	-4
0	2	6	0	4
NR1=R1-R2….NR3=R3+R2….NR4=R4-2R2
1	0	-2	2	-1
0	1	3	-1	3
0	0	0	1	-1
0	0	0	2	-2
NR1=R1-2R3….NR2=R2+R3….NR3=R3….NR4=R4-2R3
1	0	-2	0	1
0	1	3	0	2
0	0	0	1	-1
0	0	0	0	0
WE FIND THAT LAST ROW IS ALL ZEROS .
SO W SPANS ONLY R3
SO WE NEED 3 AND ONLY 3 L.I. VECTORS TO FORM ITS BASIS
WE FIND THAT V1 , V2 , V4 ARE L.I. ..
SO THE BASIS IS …..B = [V1 , V2 , V4 ] …THAT IS …
V1=	V2=	V4=
1	1	1
1	2	0
2	1	4
-1	1	-1
ANSWER…………