Let W be the subspace of R^4 defined by W = Span(1, 0, 1, 0),(1, 1, 1, 1), (0, 1
ID: 2939616 • Letter: L
Question
Let W be the subspace of R^4 defined by W = Span(1, 0, 1, 0),(1, 1, 1, 1), (0, 1, 1, 1), and
let S be the subset of R^4 given by S = (1, 0, 1, 0), (0, 1, 0, 1),(2, 3, 2, 3).
a) Show that S spans W
Solutions provided:
We can express elements of S as linear combinations of the vectors(1, 0, 1, 0), (1, 1, 1, 1),
(0, 1, 1, 1), and thus S spans W:
(1, 0, 1, 0) = 1(1, 0, 1, 0)
(0, 1, 0, 1) = (1, 1, 1, 1) -(1, 0, 1, 0)
(2, 3, 2, 3) = 3(1, 1, 1, 1) - (1, 0, 1, 0)
I understand that (2, 3, 2, 3) = 3(1, 1, 1, 1) - (1, 0, 1, 0)is based on the equation w=k1v1+k2v2...
but what if v1,v2 and v3 aren't multiples of w, lets say insteadof (1,1,1,1), it is (1,0,0,1)???
Kindly explain...
Explanation / Answer
Let W be the subspace of R^4 defined by W = Span(1, 0, 1, 0),(1, 1, 1, 1), (0, 1, 1, 1), and
let S be the subset of R^4 given by S = (1, 0, 1, 0), (0, 1, 0, 1),(2, 3, 2, 3).
a) Show that S spans W
Solutions provided:
We can express elements of S as linear combinations of the vectors(1, 0, 1, 0), (1, 1, 1, 1),
(0, 1, 1, 1), and thus S spans W:
(1, 0, 1, 0) = 1(1, 0, 1, 0)
(0, 1, 0, 1) = (1, 1, 1, 1) -(1, 0, 1, 0)
(2, 3, 2, 3) = 3(1, 1, 1, 1) - (1, 0, 1, 0)
I understand that (2, 3, 2, 3) = 3(1, 1, 1, 1) - (1, 0, 1, 0)is based on the equation w=k1v1+k2v2...
but what if v1,v2 and v3 aren't multiples of w, lets say insteadof (1,1,1,1), it is (1,0,0,1)???
but what if v1,v2 and v3 aren't multiples of w, lets say insteadof (1,1,1,1), it is (1,0,0,1)???
BUT IT SHOULD BE WHETHER
W=[W1,W2,W3] WHERE ...W1= (1, 0, 1, 0), W2=(1, 0, 0, 1), W3=(0,1, 1, 1)
CAN BE WRITTEN AS A LINEAR COMBINATION OF
S = [V1,V2,V3] WHERE
V1=(1, 0, 1, 0),V2= (0, 1, 0, 1),V3= (2, 3, 2, 3)..
NOW IF
W = Span(1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 1) AS YOU SAY THENALSO WE CAN SHOW THAT
W=[W1,W2,W3] WHERE ...W1= (1, 0, 1, 0), W2=(1, 0, 0, 1), W3=(0,1, 1, 1)
CAN BE WRITTEN AS A LINEAR COMBINATION OF
S = [V1,V2,V3] WHERE
V1=(1, 0, 1, 0),V2= (0, 1, 0, 1),V3= (2, 3, 2, 3)....SO IT ISOK
HOPE IT IS CLEAR
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