Let A be an n n matrix Is H x R n Ax 2x a subspace of Rn

Let A be an n × n matrix. Is H = {x R n | Ax = 2x} a subspace of R^n ? Which conditions for a subspace are fulfilled by H?

Solution

1st Part:

Rⁿ be a vector space over the field R. We need to check whether H = {x Rⁿ | Ax = 2x} a subspace of Rⁿ

Let x, y H and c R, c 0

Hence, x, y Rⁿ and

Ax = 2x --------------------------- (1)

Ay = 2y --------------------------- (2)

(1) + (2) A(x + y) = 2(x + y) --------------------------- (3)

Since Rⁿ be a vector space over the field R, x + y Rⁿ and cx Rⁿ

Therefore, equation (3) implies that x + y H.

Now, multiplying c both sides of equation (1) we get, c(Ax) = c(2x)

Or, Acx = 2cx (as c is a scaler, c R, c 0)

Or, A(cx) = 2(cx) --------------------------- (4)

Therefore, equation (3) implies that cx H.        

For x, y H and c R, c 0; we get x + y H and cx H.

This implies that H is a subspace of Rⁿ.

2nd Part:

Conditions for a non-empty subset W of a vector space V over a field F to be a subspace of V.

Let V be a vector space over a field F and W be a non-empty subset of V. Then W is said to be a subspace of V if the following conditions are satisfied:

As we have shown above that H (subset of Rⁿ) satisfies these conditions (for x, y H and c R, c 0; we get x + y H and cx H) to be a subspace of Rⁿ.