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# [問題] a math questions

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## a math questions

(1) (10pts) Let V and W be two (real) vector spaces and T : V →
W is a linear transformation. Suppose Z is a subspace of W,
namely Z ≤ W. Let T^−1
(Z) be the set of all vectors x in V
such that T(x) lies in Z. Namely
T^−1 (Z) = {x in V | T(x) lies in Z.}.
Argue that T^−1(Z) is also a subspace of V .
(2) Again let T : V → W be a linear transformation between two
real vector spaces V, W.
• (5pts) Suppose {v1, · · · , vp} is a linearly dependent set of
vectors in V , then the set of their images
{T(v1), · · · , T(vp)}
is also linearly dependent.
• (5pts) Suppose T is one-to-one (namely injective) and {v1, · · · , vp}
is a set of vectors in V such that
{T(v1), · · · , T(vp)}
is linearly independent (set of vectors in W), then{v1, · · · , vp}
is also linearly independent.

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