# 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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### Can set of integers form a vector space over field of rationals?

As field of reals $\mathbb{R}$ can be made a vector space over field of complex numbers $\mathbb{C}$ but not in the usual way. In the same way can we make the ring of integers $\mathbb{Z}$ as a ...
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### Proof that $\mathbb{R}^+$ is a vector space

I was doing some beginner linear algebra tasks and stumbled upon this one: Proove that $\mathbb{R}^+$ is a vector space over field $\mathbb{R}$ with binary operations defined as $a+b = ab$ (where $ab$...
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### How do we show every linear transformation which is not bijective is the difference of bijective linear transforms?

I have been reviewing some ideas about vector spaces and came upon a surprising fact. I am not quite sure how to begin the argument because the problem requires one to construct two bijective linear ...
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### Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
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### $x,y$ are linearly depending iff $|\langle x,y\rangle|=||x|| \cdot ||y||$

I tried to prove a special case of Cauchy-Schwarz: $$x,y \text{ are linearly depending vectors} \Leftrightarrow |\langle x,y\rangle|=||x|| \cdot ||y||$$ $\Rightarrow$ is simple: \begin{eqnarray*} x= ...
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### Is this proof of Cauchy Schwarz inequality circular or valid?

I'm a college freshman learning linear algebra on my own, and I'm in the section on inner products. I noticed a proof of the Cauchy Schwarz inequality for vectors in my book, and it seems to contain ...
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### Question about Normed vector space.

Here is the definition of a normed vector space my book uses: And here is a remark I do not understand: I do not understand that a sequence can converge to a vector in one norm, and not the ...
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### Linear and Commutative function over Square Matrices.

Find all functions $f$, such that $f(mA+nB) = mf(A) + nf(B)$ and $f(AB) = f(BA)$ , where $A, B$ are square matrices and $m,n$ are scalars. Need to find $f$ as an explicit function of any general ...
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### Proof of a direct sum decomposition

I was trying to prove this statement: If $N: V \to V$ is a nilpotent operator on a complex vector space, $N^k=0$ and $U\subset V$ is a subspace with $U \cap \ker(N^{k-1})= \{0\}$ then there exists ...
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### Alternative Almost Complex Structures

Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector ...
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### Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but not ...
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### Why are inner product spaces only defined on $\Bbb R$ or $\Bbb C$?

A vector space $V$ makes sense over any field $F$, or even a division ring. So why does adding an inner product suddenly not make sense without taking the $F=\Bbb R$ or $\Bbb C$? What are the primary ...
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### Given a vector space with two inner products, there is a linear transformation taking one to another

I am looking for some hint to the following question: Let $V$ be an $n$-dimensional real inner product space and let $\langle x,y\rangle$ and $[x,y]$ both be two different inner products on V. ...
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### Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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### Dimension of the vector space of homogeneous polynomials

Let $k[X_0, X_1, \ldots, X_n]_d$, or briefly $k[X]_d$, be the $k$-vector space whose elements are the zero polynomial and homogeneous polynomials of degree $d\geq 1$. I found the following formula for ...
If we were given the vector $(1,1,1)$, say, we know immediately that it belongs to the vector space $\mathbb{R}^3$ (and infinitely many others). But, if we take this vector of dolphins: or some ...