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5 votes
Accepted

Nature of the Euclidean Norm

This is subtle. In mathematics we have the freedom to define distances however we want depending on the application, and for example sometimes rather than the Euclidean norm we use what are called the ...
Qiaochu Yuan's user avatar
2 votes

Is there a finite dimensional vector space over a finite field with exactly two bases?

To give this question an answer: Any $\mathbb{F}_3$-vector space $V$ of dimension $1$ has exactly two bases, namely the two singletons $\{x\}$ and $\{y\}$, where $x$ and $y$ are the two non-zero ...
azimut's user avatar
  • 23k
2 votes
Accepted

Understanding the implication in linear algebra regarding vectors

The proposed statement reads: If for every $\mathbf{v} \in V$ it holds that $\mathbf{z} \cdot \mathbf{v} = \mathbf{w} \cdot \mathbf{v}$, then $\mathbf{z} = \mathbf{w}.$ This can be rewritten as ...
JAG131's user avatar
  • 907
2 votes

Understanding the implication in linear algebra regarding vectors

It is the wrong approach, because inserting $\mathbf{v} = \mathbf{0}$ is the not the given condition on which the conclusion is based. The condition is that for all $\mathbf{v}$, we have $\mathbf{v \...
Quadrics's user avatar
  • 24k
2 votes

Understanding the implication in linear algebra regarding vectors

Let $\mathbf{v}=(x_1,x_2,\cdots,x_n) \in V$ be any vector such that $\mathbf{z} \cdot \mathbf{v} = \mathbf{w} \cdot \mathbf{v}$. Writing, $$ \mathbf{z}=(z_1,z_2,\cdots,z_n) \quad \text{and} \quad \...
Guilherme's user avatar
  • 1,635
1 vote

direct sum of general linear space

Consider $v - u = w$. On one hand, we know that $w \in U_3$; on the other we must have that $v - u \in U_2$ since both $v$ and $u$ are in $U_2$. By assumption, $U_2 \cap U_3 = \{0\}$, so we have $w=0$ ...
Shyam R.'s user avatar
  • 171
1 vote

Affine Map as a Morphism of Affine Vector Spaces

ORIGINAL ANSWER REPLACED BY: $\newcommand\vcal{\mathcal{V}} \newcommand\acal{\mathcal{A}} \newcommand\F{\mathbb{F}} $ Let $\vcal$ be the category of vector spaces over the field $\F$ of real or ...
Deane's user avatar
  • 8,317
1 vote

Affine Map as a Morphism of Affine Vector Spaces

There are a couple ways to do this. Personally I find it unsatisfying to think of affine spaces in terms of their associated vector spaces: it is actually possible to give an axiomatization of affine ...
Qiaochu Yuan's user avatar
1 vote

Understanding the implication in linear algebra regarding vectors

If for every $\mathbf{v} \in V$ it holds that $\mathbf{z} \cdot \mathbf{v} = \mathbf{w} \cdot \mathbf{v}$, then $\mathbf{z} = \mathbf{w}$. In the syntax of mathematical logic, we can write this ...
David K's user avatar
  • 99.7k
1 vote

Nature of the Euclidean Norm

The definition is partly "an greed-upon convention", but at the same tame there is a reason to define it the way it is. There is mathematical notion called "metric" or "...
César VB's user avatar
  • 495

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