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# 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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### Is F a vector space over F?

I know $F^n$ is a vector space over F (where F is a field) This statement is true even when n is 1 right? My professor said otherwise and that got me to ask the question here, please do excuse the ...
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### Need help with part b [closed]

b) Find a basis for orthogonal complement of S, S ⊥. What is the dimension of S ⊥?
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### Is nonlinear transform of a vector space a connected set

Consider $b_i=(\theta_p-\theta_q)\sin((\theta_p-\theta_q)\alpha)$ where $p,q\in\{1,\cdots,w\}$ and $N=\frac{w(w-1)}{2}$. In order to guarantee the one-to-one correspondence between $(\theta,\alpha)$ ...
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### Calculating a weighted midpoint between two 3D (XYZ) magnitude direction vectors

I'm really struggling. I have two "3D direction/motion magnitude" vectors for a 3D game engine. The vectors consist of XYZ 3D components in the range <...
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### linear combination - finite, infinite countable, and continuous

I am a beginner student of functional analysis. We learn that, if $X$ is a vector space over $\mathbb{F}$ of finite dimension, it means it can be generated from a finite base, $V \subset X$, which ...
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### Intersection of 3 planes with linearly independent normals

In Tom Apostol's Calculus, vol. $1$, exercise $13.17.16$ is: Prove that three planes whose normals are linearly independent intersect in one and only one point. We know that every $n$ linearly ...
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### Can a vector with unordered components exist?

It seems like in order for a vector addition to be commutative, it needs to be defined in a "regular" manner, i.e. by adding matching vector components (because then the commutativity of ...
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### How to show vectors span a set?

Say if I have $V=${$(1,0,1),(0,0,1),(0,1,0)$} , how can I show that $V$ spans $\mathbb R^3$? I believe there is a theorem that if the dimension of the vector space is $n$, then $n$ linearly ...
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### Obtain the form and range of a matrix from its null space and left null space

The matrix 𝐴 has 𝑁(𝐴) [ 1 0 −1], and 𝑁(𝐴𝑇) [1 1 1 1] and [ 1 1 −1 −1]. What is the form of matrix 𝐴 and its range? I known that the null space is related to columns and left null space is ...
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### How to construct a matrix given column and null spaces?

The problem says: If possible, construct a matrix whose column space contains [1 1 0] and [0 1 1] and whose null space contains [1 0 1] and [0,0,1]. I know that the column space has to be part of the ...
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