# 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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### Plotting a 3D graph from explicit equation

I´m a 2nd year engineering student and today we learned how to plot 3d graphs from a $XYZ$ equation on paper. For example, I know ($\frac{X^2}{9}+ \frac{Y^2}{16} + \frac{Z^2}{9} =1$) will produce an ...
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### Sandwich rule for Lie algebras

On an infinite dimensional vector space an operator can be onto but not one-to-one (and vice versa). This arises the following question. Let $L_1$ and $L_2$ be Lie algebras (infinite dimensional, over ...
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### Endomorphism is normal and idempotent iff it is an orthogonal projection.

I've searched for answers for this question here for some time but haven't found an applicable answer because I could only find related questions, but not this one in particular. Suppose $V$ is a ...
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### How do I compute the gradient of a tensor?

From this paper, we have three matrices $U\in \mathbb{R}^{n\times d_U}$, $M\in \mathbb{R}^{m\times d_m}$, $C\in \mathbb{R}^{c\times d_C}$ and a tensor $S\in \mathbb{R}^{d_U \times d_M \times d_C}$, ...
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### Is there a name for a $k$-sphere embedded in $\mathbb{R}^n$?

In my thesis in a lot of places there comes up a $k$-sphere embedded in $\mathbb{R}^n$. We call lower-dimensional "planes" in $\mathbb{R}^n$ linear manifolds or flats, is there also a term for a lower-...
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### How to find a basis for subspace of functions

I am doing this exercise: The cosine space $F_3$ contains all combinations $y(x) = A \cos x + B \cos 2x + C \cos 3x$. Find a basis for the subspace that has $y(0) = 0$. I am unsure on how to ...
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### Rule for $\langle x,y\rangle$ if we know orthonormal base?

How to define $\langle x,y \rangle$ in space of polinoms, where $1, x-1 , 1-x^2$ are orthonormal base($\Vert a\Vert = 1$, $\langle a1, a2\rangle = 0$)? I'm a bit lost, I know how to do it with my ...
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### Question about Column space and Row space of generic matrix $A$ and the corresponding upper triangular $U$

I am doing the following exercise from Introduction to Linear Algebra: Find the dimensions of (a) the column space of A, (b) the column space of U , (c) the row space of A, (d) the row space of U ....
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### A possible generalized determinant?

This will likely seem a bit contrived, and admittedly it is, but I wanted to see just how "close" we could get to generalizing the concept of a determinant. In what follows, we will lose quite a few ...
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### Transformation matrix is jordan normal form

I have the following question: Given a finite-dimensional, unitary vector space V and a endomorphism f on V, is it possible to choose an orthonormal basis B of V in such a way, that the transformation ...
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### Is span $\{[1,0],[0,1]\}$ a vector space?

I can't figure this out. I would think that it is a vector space because it has the zero vector, and it seems to me to be closed under addition and scalar multiplication. But $[1,0]+[0,1] = [1,1]$ ...
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### Show that $f(y)=0$ for every $y \in Y$.

If $Y$ is a subspace of a vector space $X$ and $f$ is a linear functional on $X$ such that $f(Y)$ is not the whole scalar field of $X$, show that $f(y)=0$ for all $y \in Y$. Suppose to the contrary, ...
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### Set of all $2*2$ matrix on $\mathbb Q$ is not a vector space over $\mathbb R$

why $V= \{\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}|a,b,c,d \in \Bbb Q\}$ is not a vector space over $\Bbb R$ under usual matrix addition and scalar ...
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### Is it okay to think of functions as of vectors with “uncountable index”

In some applied areas that have a little scent of functional analysis (e.g., getting error bound in numerical methods), it is somewhat appealing for me to think of functions $\mathbb R \to \mathbb R$ ...