(a) Show that this

is a linearly independent subset of R3.
(b) Show that 0

is in the span of S by finding c1 and c2 giving a linear relationship.

Show that the pair c1, c2 is unique.
(c) Assume that S is a subset of a vector space and that is in [S], so that is a linear combination of vectors from S. Prove that if S is linearly independent then a linear combination of vectors from S adding to is unique (that is, unique up to reordering and adding or taking away terms of the form 0 ˆ™ ). Thus S as a spanning set is minimal in this strong sense: each vector in [S] is a combination of elements of S a minimum number of times-only once.
(d) Prove that it can happen when S is not linearly independent that distinct linear combinations sum to the same vector.

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