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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Axler "Linear Algebra Done Right" Exercise 6.B.13
This exercise appears in Section 6.B "Orthonormal Bases" in Linear Algebra Done Right by Sheldon Axler. Inner product spaces, norms, orthogonality, and orthonormal bases have been ...
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Is there a counter example to disprove the following regarding vector addition in binary field?
Let $\{\mathbf{a}_1 , \mathbf{a}_2 , \mathbf{a}_3 , ...., \mathbf{a}_{30}\}\subset \mathbb{F}_2^{15}$ denote the set of binary vectors. Define the set of integers $\{p_k\}_{k=1}^{14}$ as$3 \leq p_1 &...
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Is piecewise linear function necessarily convex?
In one of his lectures on convex functions, Stephen Boyd claimed piecewise linear functions are convex because a piecewise linear function can be thought of as pointwise maximum of a set of affine ...
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The $8$ laws of the definition of vector spaces are not independent. What is the relationship among them? [duplicate]
I am learning linear algebra, and have a question about the eight laws in the definition of a vector space.
A non-empty set $V$ is a vector space on field $F$, if addition $+:V\times V\to V$ and ...
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Questions about how to show $d_1+\cdots +d_n-n+1 \leq {\text{dim}}_k k[x_1,\ldots,x_n]/\mathfrak{a}\leq d_1d_2\cdots d_n\quad $
The following are from Froberg's "Introduction to Grobner bases" , and Hungerford's undergraduate "Abstract Algebra" text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\...
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What is the connection between bilinear and quadratic forms.
I know that a bilinear form $B$ on the $\mathbb R$-vector space $\mathbb R^n$ is defined to be a map $B:\mathbb R^n\times \mathbb R^n\to \mathbb R$ which is linear in each coordinate.We know that a ...
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Showing that $k[x_1,\ldots,x_n]/\mathfrak{a}$ is a finite dimensional vector space over $k$ assuming basic linear algebra and min amount of abs alg.
The following are from Froberg's Introduction to Grobner bases, and Hungerford's undergraduate Abstract Algebra text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\backsimeq (k[x_1,\ldots,x_{n-1}])...
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Show that $P_n(F)$ is generated by $\{1, x, \dots, x^n \}$. Differentiating between $\text{span}(S) \subseteq W$ and $\text{span}(S) = W$?
Show that $P_n(F)$ is generated by $\{1, x, \dots, x^n \}$.
My Work
$$S = \{1, x, \dots, x^n\}$$
$$W = \{ a_n w^n + a_{n - 1} w^{n - 1} + \dots + a_0 : a \in F \}$$
Let $x \in \text{span}(S)$.
...
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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 $\{v_1,v_2,......
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Constructing the interval [0, 1) via inverse powers of 2
If $x$ is rational and in the interval ${[0,1)}$, is it always possible to find constants $a_1, a_2, ..., a_n\in\{-1, 0, 1\}$ such that for some integer $n\geq{1}$, $x = a_1\cdot2^{-1} + a_2\cdot{2^{-...
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Applying vector decomposition multiple times and RH orthonormal bases
I want to show that I can write any 3D vector $v$ in components with respect to the right handed orthonormal basis $\{e_1, e_2, e_3\}$ (i.e. three perpendicular unit vectors $\{e_1, e_2, e_3\}$ such ...
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Basis of column/row space of $A$: using pivot columns of $A$ vs. $\text{rref}(A)$?
When we have column vectors and want to check which ones are linearly dependent to take them out and form a basis for the column space of $A$, we put them as column vectors in the matrix. Then, we ...
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How to prove that all solutions to $Ax=b$ are $(x_0 + bv_1 +cv_2)$, where $A$ is a linear transformation from $\mathbb{R}^{N}$ to $\mathbb{R}^{N}$?
The linear transformation $T:\mathbb{R}^N \to \mathbb{R}^N$ and is represented by the matrix $A$. A basis for the null space of $A$ consists of the vectors $v_1$ and $v_2$. Prove that if a particular ...
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Is this a valid vector space? (Question Verification) [closed]
Hi - please see the question above. I have a problem with proving associativity i.e. that $(\alpha \beta)v = \alpha(\beta v)$ where $\alpha, \beta \in \mathbb{R}$ and $v \in \mathbb{R}_+$.
I think ...
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Distance between subspaces after premultiplication by diagonal matrice
Let $\mathcal{E}_1$ and $\mathcal{E}_2$ be two k dimensional subspaces in $\mathbb{R}^n$ and two $n\times K$ matrices $E_1$,$E_2$ are basis matrix. Distance between two subspaces is defined to be the ...
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