Matrices by Shmuel Friedland

Author:Shmuel Friedland
Language: eng
Format: epub
Tags: Linear Algebra,

iii. The submatrix A" _fe_1 -B[;, {,]{,..., J t c }] has rank t if and

only if not all the determinants det A™"' £ " 1 B[{ii} c ,..., {i t } c }, {{Jf,..., J t c }] for 1 < ^i < *2 < « — k are equal to zero.

iv. y(J x ), ■ ■ ■ ,x(J„_fe) is a basis in the null space of A if and only if the determinant of the full row submatrix of A n_fe_1 i? corresponding to the columns determined by Jf,..., J^_ k _ 1 is not equal to zero.

10. Let C € jpnx(n-fe) k e a matrix of rank n — k. Show that there exists A e F fcx ™ of rank fc such that AC = 0.

11. Let F be a field of 0 characteristic. Let p(x\,..., x n ) £ ¥[xi,..., x n ]. Show that p(xi, ..., x n ) = 0 for all x = (i n ..., x n ) T G F" if and only if p is the zero polynomial. Hint: Use induction.

12. Let F be a field of 0 characteristic. Assume that V = F™. Identify V with F". So for u e V, f e V (u, f) = f T v. Show

(a) U C V is a subspace of dimension of n — 1 if and only if there exists a nontrivial linear polynomial Z(x) = a x x x + ... + a n x n such that U is the zero set of Z(x), i.e. U = Z(l).

(b) Let Uu ... ,Ufe be k subspaces of V of dimension n — 1. Show that there exists a nontrivial polynomial p — Yii=i h & ¥[x\,..., x n ], where each ^ is a nonzero linear polynomial, such that Uj = iUj

CHAPTER 5. ELEMENTS OF MULTILINEAR ALGEBRA

(c) Show that if ,..., Ufc are k strict subspaces of V then U^ =1 Uj is a strict subset of V. Hint: One can assume that dim Uj = n — 1, i = l,..., k and the use Problem 11.

(d) Let U, U 1; ..., Ufe be subspaces ofV. Assume that U C u£ =1 U;. Show that there exists a subspace U, which contains U. Hint: Observe U = uf =1 (U, n U).

13. Let the assumptions of Problem 12 hold. Let X = [xij], U = [uij] G

F nx/ . View the matrices A l X,A l U as column vectors in F("). Let per(a:ii,..., x n i) := det (X T U) = (A l X) T A 1 U. View p v as a polynomial in nl variables with coefficients in F. Show

(a) pu a homogeneous multilinear polynomial of degree /.

(b) Pu is a trivial polynomial if and only if rank U < 1.

(c) Let X e Gr;(V), U e Gr;(V) and assume that the column space of X = [xij] = [x 1; ... ,x/], U = [uij] = [u lr ..,U(] e F" x/ are X, U respectively.

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