Verify that each is a vector space by checking the condition

Verify that each is a vector space by checking the conditions. The collection of 2multiple2 matrices with 0\'s in the upper right and lower left entries. The collection {a_0+a_1x+a_3x^3|a_0,a_1,a_3 epsilon R} of cubic polynomials with no quadratic term.

Solution

1(a) If any two such matrices are added, the result would be a 2x2 matrix with 0\'s in the upper right and lower left entries. Hence the closure property of addition is satisfied.

Next, if the given matrix is multiplied by a scalar, the result would still be a 2x2 matrix with 0\'s in the upper right and lower left entries. Hence the closure property under scalar multiplication is also satisfied.

Hence it is verified that collection of 2x2 matrices with 0\'s in the upper right and lower left entries is a vector space.

1(b) The given cubic polynomial with no quadratic term has coefficients that are in R. Suppose we add two cubic polynomials a₀ +a₁x +a₃ x³   and b₀ +b₁x -a₃ x^{3 ,} their addition would be a polynomial (a₀ +b₀) +(a₁ +b₁)x, which is not a cubic polynomial. Hence the closure property of addition is not satisfied.

Hence the collection of cubic polynomials is not a vector space,

   ******