Please if you can send me the procedure of the problems thanks i appreciated In

Question

Please if you can send me the procedure of the problems thanks i appreciated

In the following, W_1, W_2, and W_3 are subsets of R^2 with the usual operations of vector addition and scalar multiplication. Determine whether W_1, W_2, and W_3 are subspaces of R^2. In each ease, give a formal proof or give a counterexample. W_1 is the set of all vectors of the form [x y]^T, where x greaterthanorequalto 0. W_2 is the set of all vectors of the form [x y]^T, where x greaterthanorequalto 0 and y greaterthanorequalto 0. W_3 is the set of all vectors of the form (x y)^T, where x = 0.

Explanation / Answer

For W to be subspace of R^2; property of Addition and scalar multiplication must follow in each example;

a)

W1 set of all vectors for (x y) where x>=0;

now (x1 y1) and (x2 y2) be vectors of W1; that means x1 and x2 >=0;

(x1+x2 y1+y2) that means x1+x2 >=0 which belong to W1;

also k(x1 y1) = (kx1 ky1) where x1 >=0; but kx1 not >=0 because k can be negative;

so W1 is not a subspace of R^2;

b)

W2 set of all vectors for (x y) where x>=0 and y>=0;

now (x1 y1) and (x2 y2) be vectors of W2; that means x1, x2, y1 and y2 >=0;

(x1+x2 y1+y2) that means x1+x2 >=0 and y1+y2 >=0 which belong to W2;

also k(x1 y1) = (kx1 ky1) where x1 >=0; but kx1 and ky1 not >=0 because k can be negative;

so W2 is not a subspace of R^2;

c)

W3 set of all vectors for (x y) where x=0;

now (x1 y1) and (x2 y2) be vectors of W3; that means x1 and x2 =0;

(x1+x2 y1+y2) that means x1+x2 =0 which belong to W3;

also k(x1 y1) = (kx1 ky1) where x1 =0; so that kx1 =0 which belong to W3

so W3 is a subspace of R^2;

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