Math 560, Assignment 1
Due Wednesday, September 7
1. Prove
Theorem 1.2.4
All bases for a finite dimensional linear space have the same number of elements.
2. Prove
Theorem 1.2.9
(1) Every finite dimensional vector space over Kis isomorphic to Kn,n= dim X.
(2) Any two finite dimensional vector spaces over the same field and with the same dimension are isomorphic.
3. Prove
Theorem 1.2.8
Suppose Xis a finite dimensional linear space and Yand Zare subspaces of Xsuch that X=Y+Z. Let
W=Y∩Z. Then,
dim X= dim Y+ dim Z−dim W. (1.2.1)
4. Using Definition 1.2.9, prove
Theorem 1.2.10
Let X1and X2be linear spaces over K.X1⊕X2is a linear space and
dim X1⊕X2= dim X1+ dim X2.
5. In reference to Definition 1.3.1, prove
Theorem 1.3.1
≡is an equivalence relation, i.e.,
x1≡x2⇒x2≡x1(i)
x≡x(ii)
x1≡x2, x2≡x3⇒x1≡x3(iii)
6. Prove
Theorem 1.3.3
The definitions of addition and multiplication in Definition 1.3.2 are independent of the particular choice of
representatives from the congruence classes.
7. In reference to Definition 1.2.9, prove that if Xis a linear space and Yis a subspace, then Y⊕X/Y is isomorphic
to X.
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