Let f V > W be a linear map of vector spaces over K. Prove the following (if a statement is wrong, make a correct statement and prove it): un V, if f(u),f(u2), . ,f(un) independent (a) For any U1, U2 linearly independent, then are un are (b) For any ui, U2,. .. , Un E V, if u1, U2, - linearly independent, un are then f(u), f(u2),... ,f(un) independent (c) Each vector v E K5 defines a linear functional v* : K5 >K by are 5 v" () (v, )} for x Ε Κ*. V;Xi E K Let W Row(A) and define W v (K) : v" (u) = 0 for all u W Get a basis for W
Let f V > W be a linear map of vector spaces over K. Prove the following (if a statement is wrong, make a correct statement and prove it): un V, if f(u),f(u2), . ,f(un) independent (a) For any U1, U2 linearly independent, then are un are (b) For any ui, U2,. .. , Un E V, if u1, U2, - linearly independent, un are then f(u), f(u2),... ,f(un) independent (c) Each vector v E K5 defines a linear functional v* : K5 >K by are 5 v" () (v, )} for x Ε Κ*. V;Xi E K Let W Row(A) and define W v (K) : v" (u) = 0 for all u W Get a basis for W
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let f V > W be a linear map of vector spaces over K. Prove the following (if a
statement is wrong, make a correct statement and prove it):
un V, if f(u),f(u2), . ,f(un)
independent
(a) For any U1, U2
linearly independent, then
are
un are
(b) For any ui, U2,. .. , Un E V, if u1, U2, -
linearly independent,
un are
then f(u), f(u2),... ,f(un)
independent
(c) Each vector v E K5 defines a linear functional v* : K5 >K by
are
5
v" ()
(v, )}
for x Ε Κ*.
V;Xi E K
Let W Row(A) and define
W
v
(K) : v" (u) = 0 for all u
W
Get a basis for W
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