Prove the following are not vector spaces a The set of two l

Prove the following are not vector spaces:

(a) The set of two lines in a plane that intersect (only) at the origin.

(b) The set of all points contained in the unit sphere in R3.

(c) The set of all invertible 2 x 2 matrices.

(d) The set of all polynomials of degree 2. (Note: This is different from P2 where a polynomial could be of degree 2 or less).

(e) The set of all solutions to any given second order linear non-homogeneous DE.

Solution

(a) Let S = the set of two lines in a plane that intersect (only) at the origin. Let L₁ and L₂ S. choose scalars k₁ and k₂ 0. Then L = k₁L₁ + k₂L₂ is a straight line passing through origin but it is neither equal to L₁ nor equal to L₂ as both k₁ and k₂ 0. Hence L does not belongs to S. This implies that S is not a vector space.

If we have taken S as the set all of lines in R² passing through origin, then it will be a vector space.

(b) Let S = The set of all points contained in the unit sphere in R³. Let x, y S where x = (x₁, x₂, x₃), y = (y₁, y₂, y₃); x_i, y_i R for all i = 1,2,3. Then |x_i|, |y_i| < 1 for all i = 1,2,3.

As R³ is a vector space and x, y R³, x + y R³

x + y = (x₁+ y₁, x₂+ y₂, x₃+ y₃)

|x_i| and |y_i| < 1 does not imply that |x_i+ y_i| < 1, i = 1,2,3. For example, let x = (3/4, 3/4, 3/4) and y = (7/8, 7/8, 7/8). Then both x and y S, but x + y = (3/4+7/8, 3/4+7/8, 3/4+7/8) = (13/8, 13/8, 13/8) does not belong to S as 13/8 > 1.

x, y S does not imply that x + y S

Hence, S is not a vector space.

(c) Let S = the set of all invertible 2 x 2 matrices. S is not a vector space as S does not contain the zero matrix (zero matrix is the zero element or zero vector of the vector space of all 2 x 2 matrices) . As zero matrix is not invertible, it does not belongs to S.

(d) Let S = the set of all polynomials of degree 2. S is not a vector space as S does not contain the zero polynomial. As degree of zero polynomial = 0, S does not contain zero polynomial. zero polynomial is the zero element or zero vector of the vector space P₂) Proved