Linear Algebra Answers

Suppose V is finite-dimensional and U is a subspace of V such

that dim U = dim V . Prove that U = V

Evaluate the following system of equation by Gaussian Elimination method

x1+5x2+2x3=9

x1+x2+7x3=6

-3x2+4x3=-2

Show that the given set of vectors are L.I or L.D. (i) u = [(3; 4; 0; 1); (2; -1; 3; 5); (1; 6; -8; -2)].

(ii) v = [(2; 0; 0; 7); (2; 0; 0; 8); (2; 0; 0; 9); (2; 0; 1; 0)] (iii) w = [(6; 0; -1; 3); (2; 2; 5; 0); (-4; -4; -4; -4)].

Suppose V and W are finite-dimensional and T ∈ L(V, W).

Show that with respect to each choice of bases of V and W, the matrix of T has at least dim range T nonzero entries.

Please assist.

Show that if B is Skeksymmetric n×n matrix where n is an odd positive integer, then B is not invertible

Without calculating the determinant, inspect the following:

(7,1) 1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 −2

(7.2) 1 0 0 0

0 1 0 0

0 0 0 1

0 0 1/4 0

Show that the matrix [ 4 6 6 1 −1 3 2 −5 −2 ] is not diagonalizable

a. LCompute the product AB for

A = 0 4 0 2 3 1 3 0 1 and B = 1 0 3 1 1 5 2 3 -1

b.) Use your answer in a.) to evaluate det(AB) and compare it to det(A) det(B)

c.) Determine whether or not if det(A+B) is related to det(A) + det(B).

prove that A is a square matrix, then AA^T and A + A^T are symmetric

Show that W={(x,-3x,2x)|x€R} is a subspace of R³. Also find a basis for subspace U of R³ which satisfies R³ is equal to direct sum of W and U.