Which of the following are vector spaces? Justify your...

Which of the following are vector spaces? Justify your answer! (a) The set of all row vectors of the form $(a, 3 a)$. (b) The set of all vectors of the form $(a, a+1)$. (c) The set of all continuous functions for which $f(-1)=0$. (d) The set of all periodic functions of period 1, i.e., $f(x+1)=f(x)$. (e) The set of all non-negative functions: $f(x) \geq 0$. (f) The set of all even polynomials: $p(x)=p(-x)$. (g) The set of all polynomials $p(x)$ that have $x-1$ as a factor. (h) The set of all quadratic forms $q(x, y)=a x^2+b x y+c y^2$.

Which of the following are vector spaces? Justify your answer! (a) The set of all row vectors of the form $(a, 3 a)$. (b) The set of all vectors of the form $(a, a+1)$. (c) The set of all continuous functions for which $f(-1)=0$. (d) The set of all periodic functions of period 1, i.e., $f(x+1)=f(x)$. (e) The set of all non-negative functions: $f(x) \geq 0$.
(f) The set of all even polynomials: $p(x)=p(-x)$. (g) The set of all polynomials $p(x)$ that have $x-1$ as a factor. (h) The set of all quadratic forms $q(x, y)=a x^2+b x y+c y^2$.

Key Concepts

Vector Space Definition

A vector space is a collection of objects called vectors, together with two operations: vector addition and scalar multiplication, that satisfy a specific set of axioms (such as associativity, commutativity, the existence of an additive identity and inverses, and distributive properties). This structure allows for a robust framework to perform linear combinations and other algebraic operations.

Subspace Test

A subspace is a subset of a vector space that is itself a vector space under the same operations. To verify that a set is a subspace, one must check that it is non-empty (specifically containing the zero vector), that it is closed under vector addition, and that it is closed under scalar multiplication.

Zero Vector Existence

The presence of a zero vector, which serves as the additive identity in a vector space, is crucial. Any subset of a vector space must include the zero vector to satisfy the vector space axioms, ensuring that every element has an additive inverse and that scalar multiplication behaves correctly.

Closure under Operations

For a subset of a vector space to qualify as a vector space in its own right, it must be closed under the operations of addition and scalar multiplication. This means that the sum of any two elements in the subset, and any scalar multiple of an element in the subset, must also be contained within the subset.

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