linear algebra - linearly independent?

Question

Let u , v ,w , be linearly independent vectors. A non-trivial dependence relation satisfied by the vectors 7u-224v , 8v+w , 4w+u

is _____(7u-224v)+______(8v+w)+______(4w+u) = 0

Answer 1

Respuesta preferida

h (7 u - 224v) + k (8v + w) + j (4 w +u) = 0

(7h +j) u + (-224 h + 8 k) v + (k + 4j) w = 0

Como u, v y W son L. I., resulta

7 h +j = 0

-224 h + 8 k = 0

k + 4 j = 0

I 7.......0...1 I

I -224..8...0 I

I 0........1...4 I

I 7......0.....1 I

I 0......8...32 I

I 0......1.....4 I

I7...0...1 I

I0...1...4 I

I 0...0...0 I

7h + j = 0

k + 4j = 0

k = - 4 j

j = - 7 h ------> k = 28 h

1 (7 u - 224v) + 28 (8v + w) - 4 (4 w +u) = 0