To demonstrate that $T(V)$ is isomorphic to $\mathbb{R}^3$, where $T: \mathbb{R}^4 \to \mathbb{R}^4$ is a one-to-one linear transformation, and $V$ is a vector subspace of $\mathbb{R}^4$ with $\dim(V) = 3$, we follow these steps:

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1. Define the Mapping $T$ Restricted to $V$:

The transformation $T$ maps $V$ (a subspace of $\mathbb{R}^4$) into $\mathbb{R}^4$. Specifically, we define: $T(V) = \{ T(\mathbf{v}) : \mathbf{v} \in V \}.$ Since $T$ is a linear transformation and $V$ is a subspace, $T(V)$ is also a subspace of $\mathbb{R}^4$.

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2. Show $\dim(T(V)) = \dim(V)$:

The dimension theorem for linear transformations states: $\dim(T(V)) + \dim(\ker(T|_V)) = \dim(V),$ where $T|_V$ is the restriction of $T$ to $V$. Because $T$ is one-to-one, its kernel is trivial (i.e., $\ker(T) = \{\mathbf{0}\}$). Thus: $\ker(T|_V) = \{\mathbf{0}\} \implies \dim(\ker(T|_V)) = 0.$ Substituting this into the dimension formula: $\dim(T(V)) = \dim(V).$ Since $\dim(V) = 3$, we conclude that $\dim(T(V)) = 3$.

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3. Show that $T(V)$ is Isomorphic to $\mathbb{R}^3$:

An isomorphism is a bijective linear map between two vector spaces. To prove that $T(V)$ is isomorphic to $\mathbb{R}^3$, we need to verify two conditions:

1. Linearity: The restriction $T|_V$ is linear, as it inherits linearity from $T$.
2. Bijectivity: Since $T$ is one-to-one on $\mathbb{R}^4$, its restriction $T|_V$ is also one-to-one. Additionally, $T(V) \subseteq \mathbb{R}^4$ has the same dimension as $V$, so $T|_V$ is surjective onto $T(V)$.

By these properties, $T|_V: V \to T(V)$ is a linear bijection. Since $\dim(T(V)) = 3$, and $\mathbb{R}^3$ is the unique 3-dimensional vector space (up to isomorphism), it follows that $T(V) \cong \mathbb{R}^3$.

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4. Conclusion:

The subspace $T(V)$ is isomorphic to $\mathbb{R}^3$ because:

1. $T(V)$ has the same dimension as $\mathbb{R}^3$.
2. $T|_V$ establishes a linear bijection between $V$ and $T(V)$.

Thus, $T(V)$ is isomorphic to $\mathbb{R}^3$.

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Would you like more details or examples on this topic? Here are five related questions to deepen understanding:

1. What is the role of the kernel in determining the properties of a linear transformation?
2. Can you provide an example of a linear transformation $T$ on $\mathbb{R}^4$ and compute $T(V)$ for a given subspace $V$?
3. Why is the isomorphism between vector spaces dependent only on dimension?
4. How does the concept of rank-nullity relate to the proof above?
5. Could the result differ if $T$ were not one-to-one?

Tip: For any linear transformation, understanding its kernel and image is key to analyzing its properties, such as injectivity, surjectivity, and isomorphisms.