Question

Let w be a subspace of rn, and let w ? be the set of all vectors orthogonal to w. Show that w ? is a subspace of rn using the following steps.

Answer 1

To show that w' is a subspace of Rn, we need to prove three main properties: closure under addition, closure under scalar multiplication, and the presence of the zero vector.

Step-by-step explanation:

To show that w′ is a subspace of Rn, we need to prove three main properties: closure under addition, closure under scalar multiplication, and the presence of the zero vector.

Closure under addition:

Let x and y be two vectors in W′. We need to show that x + y is also in W′. Since x and y are orthogonal to w, their sum will also be orthogonal to w.

Closure under scalar multiplication:

Let x be a vector in W′ and let c be a scalar. We need to show that cx is also in W′. Since x is orthogonal to w, scaling it by any scalar c will still result in a vector that is orthogonal to w.

Presence of zero vector:

Since w is a subspace of Rn, the zero vector 0 is in w. Since 0 is orthogonal to all vectors in w, it is also orthogonal to any vector in W′.