# Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars

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### Homeomorphic to a vector space but not itself a vector space

In Nash & Sen p.162, they show that the space of all positive definite symmetric matrices $C$, while not a vector space itself, is homeomorphic to the space of all symmetric matrices $S$ (which is ...
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### Related to Euclidean and unitary vector space

Let $V$ be a finite-dimensional Euclidean or unitary $K$-vector space. Show or refute the following statements: (i) $(f + g)^{*} = f^{*} + g^{*}$ for all $f, g ∈ \operatorname{End}(V)$ (ii) $(λf)^{*}$ ...
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### Minimizing the norm of the difference of two vectors.

Let $\mathcal{H}$ be a a vector space of finite dimension. Let $\mathcal{H_1}$ be a subspace of $\mathcal{H}$. Considering some vector $|\phi\rangle \in \mathcal{H}$ i need to show that there exists ...
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### Generalizing dot product to arbitrary order

For a vector $\mathbf{r}$, if we define $\mathbf{r}^1 \equiv \mathbf{r}$ and $\mathbf{r}^2 \equiv \mathbf{r} \cdot \mathbf{r}$, we find that $\mathbf{r}^1$ is a vector and $\mathbf{r}^2$ is a ...
Let's say I have a vector of integers such as $$\langle1,23,3\rangle \\ \langle41,5,16\rangle \\ \langle3,5,7\rangle \\ \langle10,13,31\rangle$$ and I wish to create a new vector whose entries are ...