# 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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### Proving replacement theorem?

I want to see if I am understanding the proof of the replacement theorem correctly. Let $V$ be a vector space that is spanned by a set $G$ containing $n$ vectors. Let $L \subseteq V$ be a linearly ...
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### Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
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### why symmetric matrices are diagonalizable?

A matrix is diagonalizable iff it has a set of eigen vectors which are linearly independent. Now, why is this satisfied in case of symmetric matrix ?
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### Embedding torsion-free abelian groups into $\mathbb Q^n$?

Glass' Partially Ordered Groups states without proof: Every torsion-free abelian group can be embedded into a rational vector space (as a group). Can someone link me to a proof of this? It ...
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### What should we understand from the definition of orthogonality in inner product spaces other than $\mathbb R^n$?

In the beginning of linear algebra courses, there are vectors in $\mathbb R^n$ and the dot product is introduced. We learn that if the dot product of two vectors is zero, then these vectors are called ...
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### Another proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
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### Calculate intersection of vector subspace by using gauss-algorithm

There are two vector subspaces in $R^4$. $U1 := [(3, 2, 2, 1), (3, 3, 2, 1), (2, 1, 2 ,1)]$, $U2 := [(1, 0, 4, 0), (2, 3, 2, 3), (1, 2, 0, 2)]$ My idea was to calculate the Intersection of those two ...
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### Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr 2013:...
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### Finding the Angle theta between two 2d vectors.

Find the angle α between the vectors $$\begin{bmatrix}3 \\ 5 \end{bmatrix}and \begin{bmatrix}-2 \\ 3 \end{bmatrix}$$ I found 64.654, but apparently is wrong from what the webwork says. Can anyone ...
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### First Order Language for vector spaces over fields

I'm attempting to come up with a first order language $L$ that is able to describe vector spaces over fields. I came up with a few sets of nonlogical symbols. $Rs_L=\{Scal,Vec\}$ where $Scal,Vec$ are ...
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### Is $\mathbb{R^Z}$ or its elements countable?

Continue on the self study on infinite vector spaces. According to this link, $\mathbb{R^Z}$ has elements of the following form: $$(y_k)_{k\mathbb{\in Z}}=(\dots y_{-1},y_0,y_{1}\dots)$$ Or more ...
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### $T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$

Question Let $T:V \rightarrow V$ be a linear map. How do I prove that $T \circ T = T_0$ ( the zero linear map) iff $R(T) \subset N(T)$? Attempt \begin{eqnarray} T\circ T=T(T(v))&=&T(T(v-0))\...
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### Calculating new vector positions

I'm using the following formula to calculate the new vector positions for each point selected, I loop through each point selected and get the $(X_i,Y_i,Z_i)$ values, I also get the center values of ...
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### How to find the gradient for a given discrete 3D mesh?

I have a 3D mesh that is looking like this: ie I have a set of triangles in a 3D space, and they are all linked by their edge. I have to compute the gradient associated with this field, at each edge ...
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