# 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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### Sum of all possible values for $\lambda$.

If $|\vec{a}| = 1$, $|\vec{b}| = 3$, $|\vec{c}| = 4$ and $|\vec{a}-\vec{b}|² + |\vec{b}-\vec{c}|² + |\vec{c}-\vec{a}|² ≤ 12k+λ,$ and if $k$ is prime number and $λ>0$, then what is sum of all values ...
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### Linear Algebra - question on the proof of the Replacement Theorem in Friedberg

I'm not quite fully understanding the author's use of induction to prove the replacement theorem in their book Linear Algebra: The induction hypothesis is, as I understand it, that the theorem holds ...
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### The space of bounded operators $L(X \rightarrow X)$ where $X=\ell_1$ is NOT seperable

How can one prove that the space of bounded operators $L(X \rightarrow X)$ where $X=\ell_1$ is NOT seperable? I am trying to think what can contradict seperability, but I didn't have any progress.
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### uniqueness of clutching decomposition Hatcher p.$47$ Lemma $208$

In the following accepted questions on mse (first and second) concerning K theory, in particular the uniqueness of the splitting given also in Hatcher p.$47$ Lemma $208$ is addressed by the same ...
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### How to show that the cardinality of the following set is $2^{n-1}$

Let $\mathbb{F}_2^n$ be a $n$-dimensional vector space over the finite field $\mathbb{F}_2$. Here $n>2$. Suppose that we can write $$\mathbb{F}_2^n=A_1\cup A_2,\ A_1\cap A_2=\emptyset,$$ where it ...
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### Deterministic approach to construct a special set in a vector space.

Let $\mathbb{F}_2^n$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_2$. I am looking for a deterministic approach to construct a special set $A$ in $\mathbb{F}_2^n$ such that $A$...
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### How to prove the following result related to cardinality of a particular set in a vector space? [duplicate]

Let $\mathbb{F}_2^n$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_2=\{0, 1\}$. Let $A\subset \mathbb{F}_2^n$ be its proper subset and $0\in A$. Suppose that there exists a ...
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### Confused about why having equal coefficents satisfy closure rules of vector spaces

My book says to verify the below quadratic polynomial is a subspace we take the linear combination of 2 members $$M = \{a +ax+ax^2 \mid a\ \in \Bbb{R}\} = \{(1 +x+x^2) \cdot a \mid a \in \mathbb R\}$$ ...
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### Two distinct lines for which there is no plane that contains both

So if a plane contains a line, does that mean every point of that line is a point on the plane? And if so, would lines that are skewed and perpendicular be an example of the statement?
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### Need help for understanding a sum of subspaces

I am newbie started learning the linear algebra. It might be dumb question. But I don't understand how the sum of subspace can also be subspace?! So for subset in order to be subspace, It should ...
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### What is an example of spanning set with linearly independent vectors that is not a Hamel Basis?

While trying to distinguish the definitions of Hamel and Schauder bases of infinite dimensional vector spaces, I was told that the true definition of a Hamel basis is not "a spanning set with ...
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Let $X$ and $Y$ be normed linear spaces and $T:X\to Y$ a bounded linear operator. Notation: $X'$ denote the dual space of $X$, and $T'$ the dual map of $T$. If $W\subset X$, then ${}^\circ W=\{f\in X'... • 568 1 vote 2 answers 65 views ### Is the set of functions from$[0,1]$to$\mathbb{R}$i.e.$[0,1]^{\mathbb{R}}$a vector space? While the elements of$[0,1]^{\mathbb{R}}$satisfy properties of vector spaces (commutativity, associativity etc..), I get the feeling that you can find$f,g \in [0,1]^{\mathbb{R}}$such that for some ... • 359 -2 votes 0 answers 32 views ### Question re: what makes a zero vector for vector space [closed] Question from the book Hi All, first time poster, I am confused about part c and d of the question. What does the arrow pointing to the Real number symbol mean? The answer is "constant function ... 0 votes 0 answers 25 views ### How to showing a function is square-integrable in Quantum mechanics Looking the A-1 section in Cohen-Tannoudji et al Quantum mechanics Vol. 1 book I get lost in the following:$\psi(r)=\lambda_1\psi_1(r)+\lambda_2\psi_2(r) \in F$In order to show tha$\psi(r)$is ... 1 vote 0 answers 34 views ### What is the difference between the cartesian product and direct sum of vectors? My notes give the cartesian product of the sets$X_1, . . . , X_n$as $$X_1 × · · · × X_n = \{(x_1, . . . , x_n) : x_i ∈ X_i for 1 \le i \le n\}$$ I believe we can think of a vector space$V$where$V=...
The question I am referring to is Hassani's Mathematical Physics Problem 2.18: Using the Schwarz Inequality to show that if $\{\alpha_i\}_{i=1}^{\infty}$ and $\{\beta_i\}_{i=1}^{\infty}$ is in \$\...