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| hilbert_space [2021/03/07 08:26] – [2.i.6 Orthogonal Subspaces] admin | hilbert_space [2021/04/02 19:03] (current) – [2.i.3 Subspaces] admin |
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| is a basis for it. | is a basis for it. |
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| This subspace is //effectively// the same as $\mathbb{C}^2$. If we do not bother to write down the third component then we do have a vector in $\mathbb{C}^2$ and we can always reconstruct the vector in $\mathbb{C}^3$ that it came from by just putting the zero back in the third component. Mathematicians would say that $\mathbb{C}^2$ is //**isomorphic**// to a subspace of $\mathbb{C}^3$, which means that there exists a one-to-one map between vectors in the subspace and vectors in $\mathbb{C}^3$. | This subspace is //effectively// the same as $\mathbb{C}^2$. If we do not bother to write down the third component then we do have a vector in $\mathbb{C}^2$ and we can always reconstruct the vector in $\mathbb{C}^3$ that it came from by just putting the zero back in the third component. Mathematicians would say that $\mathbb{C}^2$ is //**isomorphic**// to this subspace of $\mathbb{C}^3$, which means that there exists a one-to-one map between vectors in the subspace and vectors in $\mathbb{C}^3$. |
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| The distinction is somewhat important because $\mathbb{C}^2$ can be embedded in $\mathbb{C}^3$ in a variety of different ways. For example, The set of all vectors of the form | The distinction is somewhat important because $\mathbb{C}^2$ can be embedded in $\mathbb{C}^3$ in a variety of different ways. For example, The set of all vectors of the form |