Sage-Tips

How do you tell if an equation is a vector space?

To check that ℜℜ is a vector space use the properties of addition of functions and scalar multiplication of functions as in the previous example. ℜ{∗,⋆,#}={f:{∗,⋆,#}→ℜ}. Again, the properties of addition and scalar multiplication of functions show that this is a vector space.

How do you prove W1 and W2 are subspaces of V?

Similarly, since W1 nW2 C W2, and W2 is a subspace, we know that f + g ∈ W2 and also λf ∈ W2. So f + g ∈ W1 n W2 and also λf ∈ W1 n W2. This shows that W1 n W2 is closed under addition and scalar multiplication, so it is a subspace of V .

What is a vector space over F?

A vector space over F — a.k.a. an F-space — is a set (often denoted V ) which has a binary operation +V (vector addition) defined on it, and an operation ·F,V (scalar multiplication) defined from F × V to V . (So for any v, w ∈ V , v +V w is in V , and for any α ∈ F and v ∈ V α·F,V v ∈ V .

Does a vector space have to contain the zero vector?

Every vector space contains a zero vector. True. The existence of 0 is a requirement in the definition. Thus there can be only one vector with the properties of a zero vector.

How do you prove a vector space?

Proof. The vector space axioms ensure the existence of an element −v of V with the property that v+(−v) = 0, where 0 is the zero element of V . The identity x+v = u is satisfied when x = u+(−v), since (u + (−v)) + v = u + ((−v) + v) = u + (v + (−v)) = u + 0 = u. x = x + 0 = x + (v + (−v)) = (x + v)+(−v) = u + (−v).

Is union of two vector space is a vector space prove or disprove?

The Union of Two Subspaces is Not a Subspace in a Vector Space Let U and V be subspaces of the vector space Rn. If neither U nor V is a subset of the other, then prove that the union U∪V is not a subspace of Rn. Proof. Since U is not contained in V, there exists a vector u∈U but […]

What is a subspace of a vector space?

DEFINITIONA subspace of a vector space is a set of vectors (including 0) that satisfies two requirements: If v and w are vectors in the subspace and c is any scalar, then (i) v Cw is in the subspace and (ii) cv is in the subspace. In other words, the set of vectors is “closed” under addition v Cw and multiplication cv (and dw).

How do you find the distribution of a vector space?

Distributivity: a(u + v) = au + av and (a + b)u = au + bu for all u,v ∈ V and a,b ∈ F. Usually, a vector space over R is called a real vector space and a vector space over C is called a complex vector space.

Do all vector spaces have to obey the 8 rules?

All vector spaces have to obey the eight reasonable rules. A real vector space is a set of “vectors” together with rules for vector addition and multiplication by real numbers. The addition and the multiplication must produce vectors that are in the space.

Does every vector space have a unique additive identity?

Proposition 1. Every vector space has a unique additive identity. Proof. Suppose there are two additive identities 0 and 0′. Then 0 ′= 0+0 = 0, where the first equality holds since 0 is an identity and the second equality holds since 0′ is an identity. Hence 0 = 0′ proving that the additive identity is unique. Proposition 2.