| № |
Condition |
free/or 0.5$ |
| m96396 | Show that if A is an m x n matrix such that AAT is non-singular, then rank A = m. |
buy |
| m96397 | Show that if A is an n × n matrix, then A = S + K, where S is symmetric and K is skew symmetric. Also show that this decomposition is unique. |
buy |
| m96398 | Show that if A is an n ( n matrix, then det(AAT) ( 0. |
buy |
| m96399 | Show that if A is an orthogonal matrix, then A-1 is orthogonal? |
buy |
| m96400 | Show that if A is any m × n matrix, then Im A = A and AIn = A. |
buy |
| m96401 | Show that if A is any n × n matrix, then
(a) A + AT is symmetric.
(b) A - AT is skew symmetric. |
buy |
| m96402 | Show that if A is n ( n with n odd and skew symmetric, then det(A) = 0. |
buy |
| m96404 | Show that if An = O for some positive integer n (i.e., if A is a nilpotent matrix), then det(A) = 0. |
buy |
| m96410 | Show that if {v1, v2) is linearly independent and v3 does not belong to span {v1, v2}, then {v1, v2, v3} is linearly independent. |
buy |
| m96411 | Show that if W is a subspace of a finite-dimensional vector space V, then W is finite-dimensional and dim W ( dim V. |
buy |
| m96412 | Show that if W is a subspace of a finite-dimensional vector space V and dim W = dim V, then W = V. |
buy |
| m96413 | Show that if W is a subspace of an inner product space V that is spanned by a set of vectors S, then a vector u in V belongs to W⊥ if and only if u is orthougonal to every vector in S. |
buy |
| m96417 | Show that in R3,
(a) i ∙ i = j ∙ j = k ∙ k = 1;
(b) i ∙ j = i ∙ k = j ∙ k = 0. |
buy |
| m96419 | Show that
is similar to |
buy |
| m96420 | Show that it is not possible to construct a finite state machine that recognizes precisely those sequences in the language A = {0.1 | j ∈ +, i > j}. (Here the alphabet for A is ∑ = {0, 1}.) |
buy |
| m96421 | Show that P, the set of all polynomials, is a vector space? |
buy |
| m96431 | Show that the diagonal entries of any hermitian matrix are real. |
buy |
| m96438 | Show that the following properties hold provided that the transformations link together in such a way that all the operations are defined.
(a) R(ST) = (.RS)T
(b) 1wT = T = T1y
(c) R(S + T) = RS + RT
(d) (S + T)R = SR + TR
(e) (aS)T = a(ST) = S(aT) |
buy |
| m96441 | Show that the function L defined in Example 8 is a linear transformation. |
buy |
| m96443 | Show that the homogeneous system
(a - r)x + dy = 0
Cx + (b - r)y = 0
Has a nontrivial solution if and only if r satisfies the equation (a - r) (b - r) - cd = 0. |
buy |