(c) Verify, if the set W = {(x, y, x + pi y): x, y elementof R} is a subspace of
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Question
(c) Verify, if the set W = {(x, y, x + pi y): x, y elementof R} is a subspace of R^3 or not? (d) Verify, if the set W = {(x, y, x + pi y + 13): x, y elementof R} is a subspace of R^3 or not? (e) Verify, if the set W = {(y^2 + z^2, y, z): y, z elementof R} is a subspace of R^3 or not? (f) Verify, if the set W = {(0, y, z): y, z elementof R} is a subspace of R^3 or not? (g) Verify, if the set W = {(x y 0 x + pi y): x, y elementof R} is a subspace of M_2 times 1 or not? (h) Verify, if the set W = {(x 0 y x + pi y + 13): x, y elementof R} is a subspace of M_2 times 2 or not? (i) Verify, if the set W = {(x, y, z): x, y, z elementof R, z is an integer} is a subspace of R^3 or not? Intuitively, for W to be a subspace, each coordinate should be a homogenous linear polynomial, in some free variables, like x, y etc. On subspace of C(-1, 1) Let V = C(0, 1) be the vector space of all continuous real valued functions on the interval (-1, 1), with usual addition and scalar multiplication. (a) Verify, if the set W] {f elementof V: f(0) = 0} is a subspace of V or not?Explanation / Answer
We say that W is subspace if the following three conditions are met.
1) W is nonempty, W.
2) If pW and qW, then p+qW.
3) If aC and pW, then apW.
Ans(c):
W={(x,y,x+pi y): x,yR}
Check for first condition:
clearly it is non-empty, like you can take x=1,y=1 to get some element.
Check for second condition:
say p and q are in W then p=(x1,y1,x1+pi y1) and q=(x2,y2,x2+pi y2)
for some xi and yi in R
then p+q=(x1,y1,x1+pi y1)+(x2,y2,x2+pi y2)
p+q=((x1+x2), (y1+y2), (x1+x2) +pi(y1+y2))
we know that sum of two real numbers is also a real number so we can rewrite it as
p+q=(xk, yk, xk+ pi yk) where xk=x1+x2, yk=y1+y2
now p+q=(xk, yk, xk+ pi yk) looks clearly in form of W hence p+qW.
Check for third condition:
say p=(x1,y1,x1+pi y1) is in W and a in R
then ap=(ax1,ay1,ax1+api y1)
we know that product of two real numbers is also a real number so we can rewrite it as
ap=(xn,yn,xn+pi yn)
Now ap=(xn,yn,xn+pi yn) looks clearly in form of W hence apW.
As all the three conditions are verified hence W is a subspace of R^3
similarly you can test others.
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