3 votes
Accepted

∂/∂𝑥 and ∂/∂𝑦, are they elements of a Ring or a Vector Space??

They live inside a structure which is both a ring and a vector space in a compatible way, called an associative algebra (compatibility means multiplication is bilinear). This algebra is called the ...
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1 vote
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How do you call it when you stack vectors on top of each other?

Looking at Notation for set of concatenated vectors?, as @JMoravitz says, "concatenation" is probably the best name for it. Notationally, it's quite common to write this as $$w = \pmatrix{u\\...
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  • 58
1 vote
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Smooth convex set intersects any vector non-coplanar with supporting hyperplane?

Since $C$ has dimension $n$ it has interior points. Since $K$ is smooth at the origin, its normal cone is a ray, i.e., $N_K(0)= \mathbb R^+ u$ for some non-zero normal vector $u$, where $u\perp b_i$ ...
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  • 42.5k
1 vote

Smooth convex set intersects any vector non-coplanar with supporting hyperplane?

Suppose the surface is just locally $C^1$ and has non-empty interior. For simplicity, I will take the supporting plane to be $x_n=0$ and I will call the rest of coordinates $x'$, so a generic point ...
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  • 13.3k
1 vote

Size of linearly independent set bounded by the size of a spanning set

Off the top of my head, we know that $\dim V\le n$, since the dimension is the size of a minimal spanning set. But if $m>n$, we get $\dim V\ge m\gt n \rightarrow\leftarrow$.
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  • 104
1 vote
Accepted

Reference request for Fenchel-Rockafellar duality for dual system

Actually for a dual system it suffices $X$ and $Y$ to be linear spaces, i.e., no apriori topological structure.They get their topological structures from the weak topologies. In the presence of (...
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1 vote
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Alternative Solution to Theorem 5, Section 2.3 of Hoffman’s Linear Algebra

An explicit construction of a list of sequences $B_0,B_1,\ldots,B_n$ is given (while it is not written, $B_0=[v_1,\ldots,v_m]$ is the initial sequence. (Also sequences are written as sets, which they ...
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