In Exercises 1–16 a set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold. The set of all triples of real numbers (x, y, z) with the operations 1. (x, y, 2) + (x', y',z') = (x +x',y +y',z+z') and k(x, y,z) = (kx,y,z)

Is the set of rational numbers with the usual definitions and scalar multiplication a vector space? Please show justification for answer and explain steps.

1) Suppose v1, V2, V3, V4 span a vector space V. Prove that the list v1 - v2, v2 – v3, V3 - V4, V4 also spans V.

Show that the set of polynomials of degree 3 form a vector space. P(x) = ax³ + bx² + cx+d, a, b, c, d ER

Compute the products in Exercises 1–4 using (a) the definition, as in Example 1, and (b) the row–vector rules for computer Ax, or, the rule for computing a product Ax in which the i th entry of Ax is the sum of the products of corresponding entries from row i of A and from the vector x. If a product is undefined, explain why.

The set containing the elements specified also possesses its relevant operations of multiplication over reals and vector addition. Is the set a real vector space?

If V1, V2, V3, V4, and W are all vector spaces.  Are L(V1x...xV4, W) and L(V1, W) x...x L(V4, W) isomorphic vector spaces?

Show that the set of all real polynomials with a degree n < 3 associated with the addition of polynomials and the multiplication of polynomials by a scalar form a vector space.

The set containing the elements specified also possesses its relevant operations of multiplication over reals and vector addition. Is the set a real vector space?