Vector Spaces Calculator

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Example

Created on 2024-06-20Asked by Elijah Sanchez (Solvelet student)

Determine if the set

V = \{(x, y) \mid x, y \in \mathbb{R}\}

with the operations of vector addition and scalar multiplication forms a vector space.

Solution

To determine if the set

V = \{(x, y) \mid x, y \in \mathbb{R}\}

forms a vector space: 1. **Check the axioms of a vector space:** - **Closure under addition:** For

\mathbf{u} = (u_1, u_2)

and

\mathbf{v} = (v_1, v_2)

V

\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2)

is also in

V

. - **Closure under scalar multiplication:** For

\mathbf{u} = (u_1, u_2)

V

and

c \in \mathbb{R}

c \mathbf{u} = (cu_1, cu_2)

is also in

V

. 2. **Check other axioms:** - **Associativity of addition:**

(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})

. - **Commutativity of addition:**

\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}

. - **Existence of additive identity:** There exists

\mathbf{0} = (0, 0) \in V

such that

\mathbf{u} + \mathbf{0} = \mathbf{u}

for all

\mathbf{u} \in V

. - **Existence of additive inverse:** For every

\mathbf{u} = (u_1, u_2) \in V

, there exists

-\mathbf{u} = (-u_1, -u_2) \in V

such that

\mathbf{u} + (-\mathbf{u}) = \mathbf{0}

. - **Distributivity of scalar multiplication over vector addition:**

c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}

. - **Distributivity of scalar multiplication over scalar addition:**

(c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}

. - **Associativity of scalar multiplication:**

c(d\mathbf{u}) = (cd)\mathbf{u}

. - **Existence of multiplicative identity:**

1\mathbf{u} = \mathbf{u}

for all

\mathbf{u} \in V

. 3. **Result:** The set

V = \{(x, y) \mid x, y \in \mathbb{R}\}

with the operations of vector addition and scalar multiplication forms a vector space as it satisfies all the vector space axioms. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Lucas Jackson on Solvelet

1. Determine if the set of polynomials of degree less than or equal to 3 forms a vector space with respect to standard addition and scalar multiplication.2. Find a basis for the vector space spanned by the vectors

\langle 1, 2, 0 \rangle

and

\langle 0, 1, -1 \rangle

DefinitionA vector space is a set of vectors that can be added and multiplied by scalars, satisfying certain axioms or properties such as integration, integration, and distribution. For example: R2 represents the set of all two-dimensional vectors, which is itself a space vector with standard addition and scalar multiplication.

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