need help calc4 3 Le w be the union of the first and third q

need help calc4

3. Le w be the union of the first and third quadrant in the xy-plane. That is w a) If u is in w and c is any scalar, is cu in W? Why? b) Find specific vectors u and v in w such that u +vis not in w. lsw a subspace of R3

Solution

(A) In the first quadrant x,y are both positive whereas in the third quadrant both are negative,xy as given in the question & confirm from above statement is greater than or equal to zero.

W is the Union of the mode of variables(x,y here) in first & third quadrant means it covers all the positive number value.

Now as given in n the question if u in w then for a scalar c,cu definitely in w or we can say a subset of w because w consists of all possible value & if c is a scalar quantity then surely cu belongs to w.

OR

If u is in w

Suppose u=(x1_,y1) & it\'s locate on the plane w,means x1+y1=0

K is any scalar ,ku=kx1+ky1=k(x1+y1)=k×0

=0, 0=0

Indicates ku is in w.

(B)let vector u= x₁+y_1,v=x₂+y₂ both belong in w,

u+v is not in w( given in the question)

u+v=x₁+y₁+x₂+y₂ not belong to w,hence w is not a subspace of R²