# Tagged Questions

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

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### Are there any differences between the mathematical definition of vectors and scalars and how they are defined in physics?

From a purely mathematical perspective, the notion of scalars and vectors and their different roles makes sense to me. Vectors are elements of a given vector space $V$, and scalars are elements of the ...
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### Getting starting/endings points Related to Displacement Vector

I am using this resource to calculate the distance between two 3d line segments. At the end it provides the 3D Vector dP. The length of this vector provides the correct distance between the two lines ...
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### Why are vectors portrayed as a sub space of $R^n$

Typically I have seen all vector notation showing it as a sub space of $R^n$. Why not $Q$ or $Z$?
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### Let V subspace of W, with dimV=dimW. Why should they be equal?

The fact that $V$ is a subspace of $W$ , means $\dim V \leq \dim W$ . We are told $\dim V = \dim W.$ So if $B_1$ is the basis of $V$ and $B_2$ is the basis of $W,$ it is $|B_1|= |B_2| .$ We know that ...
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### Showing that a function in a vector space is linear

Let $X$ be a vector space and consider a function $f : X \rightarrow \mathbb{R}$ defined for some $a \in X$ defined as $f_a (x) = a \cdot x$. (i) Prove that $f_a (x) = a \cdot x$ is a linear function....
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### Two vector spaces with same dimension and same basis, are identical?

Let $V$ subspace of $W$ and both have same dimension and same basis. Then can we safely say that $V= W$ ? I believe yes. For example there may be an element $x \in V$ written as a linear combination ...
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### Find m so that $(m+1,1,1)$ , $(1,-m,-1)$ , $(m,1-m,2)$ are linearly dependent

I formed an augmented matrix $$\left(\begin{array}{ccc|c}m+1&1&m&0\\1&-m&1-m&0\\1&-1&2&0\end{array}\right)$$ I now that we do reduced row echelon form for the ...
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### Trying to visualize and understand double dual space

Currently I am reading "Finite-dimensional vector spaces" by Paul Halmos. I would have a question regarding the theorem on page 25. It says: If $V$ is a finite-dimensional vector space, then ...
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### $V$ be a vector space , $T:V \to V$ be a linear operator , then is $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$? [duplicate]

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$ ? (note that the direct product is well-defined as both the spaces ...
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### $V$ be a vector space , $T:V \to V$ be a linear operator , then is $\ker (T) \cap R(T) \cong R(T)/R(T^2)$?

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $\ker (T) \cap R(T) \cong R(T)/R(T^2)$ ( where $R(T)$ denotes the range of $T$ ) ? I know that the statement ...
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### What is the simplest example of the tame representation type?

What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ...
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### Hilbert space is orthornormality needed for representation?

In a Hilbert space $H$ with countable basis, if I know there is a countable basis $\{h_n\}$ of $H$ then can I express every element $h\in H$ therein as: h = \sum_n \langle h,h_n\...
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### Concerning the Gilbert Strang's book about algebra and the special solution of the nullspace.

Unfortunately I don't have yet 10 reputation, so I can't post the pic from the book, so I will paste the link. https://s32.postimg.org/g8divtz6t/Screen_Shot_2016_07_15_at_01_56_24.png My question is-...
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### help with vector calculus [closed]

the question is : how do I prove that: $\nabla^2 (r^n\vec r)=n(n+3)r^{n-2}\vec r$
### evaluating curl of $\vec r/r^2$
how do I calculate curl of : $\vec r/r^2$ I don't know how to solve this problem can someone help me please
### How to define a transported version of the simplex $X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$?
Let the simplex in $\mathbb{R}^n$ be denoted as $$X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$$ So it looks like: I want to take a point $\bar x \in X$ (red dot) And drag it to ...