91Ó°ÊÓ

In \(V\) prove the parallelogram law: $$ \|u+v\|^{2}+\|u-v\|^{2}=2\left(\|u\|^{2}+\|v\|^{2}\right) . $$ Explain what this means geometrically in the special case \(V=F^{(3)}\), where \(F\) is the real field, and where the inner product is the usual dot product.

If \(V\) is finite-dimensional and \(W\) is a subspace of \(V\) such that dim \(V=\) \(\operatorname{dim} W\) prove that \(V=W\).

If \(n>m\) prove that there is s homomorphism of \(F^{(n)}\) onto \(E\) with a kernel \(W\) which is isomorphic to \(F^{(n-m)}\).

If \(T\) is a homomorphism of \(M\) onto \(N\) with \(K(T)=A\), prove that \(N\) is isomorphic (as a module) to \(M / A\).

Let \(V\) be the real funetions \(y=f(x)\) satisfying \(d^{2} y / d x^{2}+9 y=0 .\) (a) Prove that \(V\) is a two-dimensional real vector space. (b) In \(V\) define \((y, z)=\int_{0}^{\pi} y z d x\). Find an orthonormal basis in \(V\).

If \(A\) and \(B\) are submodules of \(M\) prove: (a) \(A \cap B\) is a submodule of \(M\). (b) \(A+B=|a+b| a \in A, b \in B\\}\) is a submodule of \(M\). (c) \((A+B) / B\) is isomorphic to \(A /(A \cap B)\).

If \(v \neq 0 \in F^{(n)}\) prove that there is an element \(T \in\) Hom \(\left(F^{(n)}, F\right)\) such that \(v T \neq 0\).

If \(V\) is finite-dimensional and \(T\) is a homomorphism of \(V\) into itself which is not onto prove that there is some \(v \neq 0\) in \(V\) such that \(v T=0\).

Find the ranks of the following systems of homogeneous linear equations over \(F\), the field of real numbers, and find all the solutions. (a) \(x_{1}+2 x_{2}-3 x_{3}+4 x_{4}=0\) \(x_{1}+3 x_{2}-x_{3}=0\) \(6 x_{1}+x_{3}+2 x_{4}=0 .\) (b) \(x_{1}+3 x_{2}+x_{3}=0\) \(x_{1}+4 x_{2}+x_{3}=0\) (c) \(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=0\) \(x_{1}+2 x_{2}=0\) \(4 x_{1}+7 x_{2}+x_{3}+x_{4}+x_{5}=0\) \(x_{2}-x_{3}-x_{4}-x_{5}=0\)

An \(R\) -module \(M\) is said to be imeducible if its only submodules are (0) and \(M\). Prove that any unital, irreducible \(R\) -module is cyclic.