Determine the dimension of each of the following subspaces of 𝔽^4. (a) {(x1, x2, x3, x4) ∈𝔽^4 |x

step 1 For chapter 2, section 3, exercise 14. For A, we have the dimension of S equals 0. This is if X

Determine the dimension of each of the following subspaces...

Determine the dimension of each of the following subspaces of $\mathbb{F}^{4}$. (a) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=0\right\}$. (b) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=x_{1}+x_{2}\right\}$. (c) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=x_{1}+x_{2}, x_{3}=x_{1}-x_{2}\right\}$. (d) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=x_{1}+x_{2}, x_{3}=x_{1}-x_{2}, x_{3}+x_{4}=2 x_{1}\right\}$. (e) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{1}=x_{2}=x_{3}=x_{4}\right\}$.

Determine the dimension of each of the following subspaces of $\mathbb{F}^{4}$.
(a) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=0\right\}$.
(b) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=x_{1}+x_{2}\right\}$.
(c) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=x_{1}+x_{2}, x_{3}=x_{1}-x_{2}\right\}$.
(d) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=x_{1}+x_{2}, x_{3}=x_{1}-x_{2}, x_{3}+x_{4}=2 x_{1}\right\}$.
(e) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{1}=x_{2}=x_{3}=x_{4}\right\}$.

Key Concepts

Dimension

The dimension of a vector space (or subspace) is defined as the number of vectors in a basis for that space. It represents the number of independent directions or degrees of freedom within the space, and when constraints are applied through linear equations, the number of free variables typically determines the dimension.

Parametrization of Subspaces

Parametrization involves expressing the elements of a subspace in terms of free parameters. When a subspace is defined by linear equations, these equations can often be solved to express some coordinates in terms of others. The number of free parameters left after applying all constraints equals the dimension of the subspace.

Linear Constraints and Free Variables

In the context of subspaces defined by linear equations, each independent constraint typically reduces the number of free variables by one. Evaluating the independence of these constraints is crucial since only independent conditions reduce the dimension. This interplay between constraints and free variables is key to determining the structure and dimension of the subspace.

Vector Subspace

A vector subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. It must contain the zero vector and, when defined by linear equations, these defining conditions restrict the elements of the space while ensuring that the structure of a vector space is maintained.

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