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hello. this is from my 'spanning sets' lesson from vectors:
the set of vectors {(1,0,0), (0,1,0)} spans a set in R3
a. describe the set
b. write the vector (-2, 4, 0) as a linear combination of these vectors
c. explain why it is not possible to write ( 3,5,8) as a linear combination of these vectors
d. If we added the vector (1,1,0) to this set, would it now span R3? Explain.
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To solve the problem, I will introduce designations for these basic vectors
e1 = (1,0,0), e2 = (0,1,0).
(a) The spanned set in is the set of all triples (vectors/points) {(x,y,0)},
where the last third component/coordinate is 0 (zero).
You may think about this set as about a plane in defined by equation z= 0.
(b) (-2,4,0) = (-2)*e1 + 4*e2.
This equality is from the area of OBVIOUS truths.
You may check it by using the rules of multiplying a vector by a number
and the rules of addition vectors.
(c) It is not possible to write (3,5,8) as a linear combination of these given vectors
e1 and e2, because each and every such combination will have the third coordinate 0 (zero),
while the given vector (3,5,8) has the third coordinate 8 (not zero).
(d) The answer is NO.
The answer is NO, because the expanded set of basic vectors will still produce zero third coordinate
in any linear combination.
Adding this vector (1,1,0) to e1 and e2 does not change the spanned set: it remains the plane z= 0.
Solved, with all necessary explanations.