By V. V. Prasolov
This ebook includes the fundamentals of linear algebra with an emphasis on nonstandard and neat proofs of identified theorems. the various theorems of linear algebra bought more often than not in past times 30 years are typically overlooked in text-books yet are particularly obtainable for college kids majoring or minoring in arithmetic. those theorems are given with entire proofs. There are approximately 230 issues of suggestions.
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Extra info for Problems and theorems in linear algebra
Example text
E∗n does not depend on the choice of a basis in V , and only depends on the subspace W itself. Contrarywise, the linear span of the vectors e∗1 , . . , e∗k does depend on the choice of the basis e1 , . . , en ; it can be any k-dimensional subspace of V ∗ whose intersection with W ⊥ is 0. Indeed, let W1 be a k-dimensional subspace of V ∗ and W1 ∩ W ⊥ = 0. Then (W1 )⊥ is an (n − k)-dimensional subspace of V whose intersection with W is 0. Let ek+1 , . . , ek be a basis of (W1 )⊥ . Let us complement it with the help of a basis of W to a basis e1 , .
We may assume that the covectors f1 , . . , fk are linearly independent and fi = k k j=1 λij fj for i > k. , k (2) bi = λij bj for i > k. j=1 Let us prove that if conditions (2) are verified then the system (1) is consistent. Let us complement the set of covectors f1 , . . , fk to a basis and consider the dual basis e1 , . . , en . For a solution we can take x0 = b1 e1 + · · · + bk ek . The general solution of the system (1) is of the form x0 +t1 ek+1 +· · ·+tn−k en where t1 , . . , tn−k are arbitrary numbers.
Let e = eP and ε = εQ be other bases. , A = Q−1 AP is the matrix of f with respect to e and ε . The most important case is that when V = W and P = Q, in which case A = P −1 AP. Theorem. For a linear operator A the polynomial |λI − A| = λn + an−1 λn−1 + · · · + a0 does not depend on the choice of a basis. Proof. |λI − P −1 AP | = |P −1 (λI − A)P | = |P |−1 |P | · |λI − A| = |λI − A|. The polynomial p(λ) = |λI − A| = λn + an−1 λn−1 + · · · + a0 is called the characteristic polynomial of the operator A, its roots are called the eigenvalues of A.