Determine whether the given set S is a subspace of the vector space V V = R2, an

Question

Determine whether the given set S is a subspace of the vector space V V = R2, and S consists of all vectors (x1, X2) satisfying V = Mn(R), and S is the subset of all n Times n matrices with det(A) = 0. V = P2, and S is the subset of Pi consisting of all polynomials of the form p(x) = x2 + C. V = P3, and S is the subset of P3 consisting of all polynomials of the form p(x) = ax3 + bx. 1 is the vector space of all real-valued functions defined on the interval ( -infinity, infinity): and S is the subset of V consisting of those functions satisfying f(0) = 0. V = Mn(R)a and S the subset of all symmetric matrices V = C2 (I).. and S is the subset of V consisting of those functions satisfying the differential equation (y" - 4 y + 3 y = 0.

Explanation / Answer

For a set S to be a subspace of vector space V,the following conditions must be satisfied:-

i ) S must be a subset of V

ii ) S must be a vector space.

a. Let a=(x,y) belongs to S. so x2 = y2

so x=y or x=-y , now a = (x,x) or (x,-x)

   (x,x) + (x,-x) = (2x,0) -> doesnot belong to S

   S is not a subspace as it is not avector space.

b . now let a & b belongs to S, so det(a)=det(b)=0

   so ka+cb belongs to S as det(a+b)=0+0=0

ka also belongs to S as det(ks)=0 (k is a constant)

   it is a subspace of Mn

Note that u have to check 1st 2 properties of vector space for subspace determination & if any condition is not satisfied, then it's not a subspace.

Similarly, C is not a subspace, D,E,F & G are subspaces

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