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| hilbert_space [2021/03/07 07:36] – [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 |
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| A set of orthogonal subspaces $V_1,V_2,\cdots \subset V$ is said to //**span**// the inner product space $V$ if all vectors $\psi \in V$ can be written as | A set of orthogonal subspaces $V_1,V_2,\cdots \subset V$ is said to //**span**// the inner product space $V$ if all vectors $\psi \in V$ can be written as |
| \[\psi = \sum_j a_j \psi_j,\] | \[\psi = \sum_j \psi_j,\] |
| where the $a_j's$ are scalars and $\psi_j \in V_j$. We sometimes write this as $V = \oplus_j V_j$ | where $\psi_j \in V_j$. We sometimes write this as $V = \oplus_j V_j$ |
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| As an example, let $\phi_1,\phi_2,\cdots$ be an orthonormal basis for $V$ and let $V_j$ be the one dimensional subspace consisting of all vectors of the form $a\phi_j$. Then, $V = \oplus_j V_j$ just by the definition of a basis, i.e. all vectors $\psi \in V$ can be written as | As an example, let $\phi_1,\phi_2,\cdots$ be an orthonormal basis for $V$ and let $V_j$ be the one dimensional subspace consisting of all vectors of the form $a\phi_j$. Then, $V = \oplus_j V_j$ just by the definition of a basis, i.e. all vectors $\psi \in V$ can be written as |