# 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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### dimensions of two subspaces of a vector space not equal

I have a problem to find a relationship between two subspaces of a vector space. The two subspaces are $W_1$ which is the span of {$v_1,v_2,...,v_{n-1}$} and $W_2$ which is the span of ...
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### $P,Q,R$ be subspaces of a vector space $V$ such that $V=P \cup Q \cup R$ , then must one of $P,Q,R$ be equal to $V$? [duplicate]

Let $P,Q,R$ be subspaces of a vector space $V$ such that $V=P \cup Q \cup R$ , then is it true that one of $P,Q,R$ must be equal to $V$ ? I know the result about subspaces that tells that if for ...
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### Proving that the range of a linear operator $A: V \to W$ is the span of the image of a basis of $V$

Let $A: V \rightarrow W$ be a linear operator and {${v_{1}, v_{2},...,v_{n}}$} be a basis of the vector space $V$. Prove that $$Range(A) = span(Av_{1},Av_{2},...,Av_{n}).$$ Let $x_{1},x_{2} \in V$ ...
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### Linear independence in a vector space question

I'm working out the following question: If $\{u_1, u_2, u_3\}$ is a linearly independent set in some vector space. Explain why if $a_1u_1 + a_2u_2 + a_3u_3 = b_1u_1 + b_2u_2 + b_3u_3$, where ...
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### How to know if a point in a circle has crossed a plane passing through the center point?

I am creating a control in .NET which computes polar coordinates based on $(x,y)$- coordinates within a panel control. Here is an image to use as a reference: When the mouse moves over the circle, ...
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### Equivalence of norms in finite dimension over complete fields is true, but false for finite rank modules over complete rings

We know that if $k$ is complete valued field and $V$ a finite dimensional vector space then all norms on $V$ are equivalent. (The field is not necessarily of characteristic $0$ and its absolute value ...
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### Converting $\mathbb{C}$ linear tranformation with determinant $a+bi$ into an $\mathbb{R}$-linear transformation with determinant $a^2+b^2$.

Let $V=\mathbb{C}^2$. Let $T:V\rightarrow V$ denote a $\mathbb{C}$ linear tranformation with determinant $a+bi$, $a,b\in \mathbb{R}$. Prove that if we regard $V$ as a $4-$dimentional real vector ...
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### Finite dimensional vector space $V$ and $\operatorname{End}_k(V)$.

This is a homework problem. I want to solve it independently as best I can, so please only give awesome hints. Let $k$ be a field. Let $V$ be a vector space over $k$. I want to prove that $V$ is ...
### How is $\mathbb{C}\times\mathbb{C}$ a real vector space?
I'm working on Linear Algebra homework. I'm having trouble with: $\mathbb{C}\times\mathbb{C}$ is a real vector space. Explain why. Write down a basis for this real vector space. I'm just confused on ...