WA 3
This writing assignment is worth 5 (extra) grade points.
1. Which of the following subsets of the vector space C[โ1,1] of continuous
functions on the segment [โ1,1] are subspaces (justify your answers)?
(1 pt.) (a) The set Xof functions fwhich satisfy f(โ1) + f(1) = 1.
(1 pt.) (b) The set Yof functions fwhich satisfy
f(โt) = f(t) for every t โ[โ1,1].
(Such functions are called even.)
(3 pt.) 2. Let Tbe a linear transformation mapping a vector space Vonto a
vector space W. Prove that the following statements are equivalent:
(a) Tis left invertible, i.e., there exists a linear transformation Swhich satisfies
S(T(x)) = xfor every x โV;
(b) ker T={0};
(c) Tis a one-to-one transformation.
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