16. Suppose X and Y are both vector spaces over F. What does it mean to say that
ID: 3028682 • Letter: 1
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16. Suppose X and Y are both vector spaces over F. What does it mean to say that T is a Y linear transformation from X to Y? 17. State the dimension theorem, 18. Suppose x and Y are both vector spaces over F. What does it mean to say that T is an isomorphism from X to Y? 19. Suppose t and Y are vector spaces over F, with bases a (ui,..., tum) and B (vi, ...,vm) respectively. Let T be a linear transformation from x to Y Explain how to create the matrix representation for Twith respect to a and B, and be certain to use the notation that we have used for this. 20. Suppose xand Y are vector spaces over F. with bases a tu ...,un) and B (v1, ...,vm) respectively. Let Tbe alinear transformation from x to y, and suppose [T] What properties do Tand LA have in common? la 21. Supposex and Y are vector spaces over F with equal finite dimension, and let Tbe a linear transformation from xto Ythat is one to one. What can you conclude? 22. Supposexand Y are both vector spaces over F. What does it mean to say that X and Y are isomorphic? 23 Suppose X and Y are both vector spaces over F and T is a linear transformation from x to Y. What is N(T)?Explanation / Answer
16) X,Y are two vector spaces over F
T is linear transformation from X to Y means
T is a map from X to Y such that the following properties holds:
1) T(x+y)=T(x)+T(y) ; x, y in X
2) T(c.X)=c.T(x) ,c in F and x in X
17) dimension theorem :_______> for given vector space V , any two bases have the same number of cardinality .
18) X and Y are two vector spaces T is an isomorphism from X to Y means T is an one one onto linear transformation from X to Y
19)
T:X------------> Y
write the bases of X into the linear combination of bases of y
T(u1)=c11v1+c12v2+....c1mvm
T(u2)= c21v1+c22v2+....c2mvm
.
.
T(un)=cn1v1+cn2v2+....cnmvm
and matrix representation will be
c11 c21 ..... cn1
c12 c22 cn2
.
c1m c2m cnm [T]___alpha ^beta
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