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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How to find basis of vector fields?
I'm figuring out definition of vector fields over a manifold as differentiations of algebra $C^\infty(M)$ of functions on $M$. How can we find their basis starting from this very definition? I know, ...
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Proving the Set of Periodic Functions with Restrictions Form a Vector Space
I understand that a set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ cannot be a vector space because the set is not closed under the sum of the functions, as discussed here. However, I ...
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Graphical Intuition of a Linear Transformation in terms of Row Vectors
The graphical intuition of a linear transformation (matrix) $A \in \mathbb{R}^{m \times n}$ applied on a vector $\textbf{v}$ in terms of the column vectors $\textbf{c}_i$ of $A$ is quite clear to me:
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Stability of Subspaces under a Linear Map in Direct Sum Decomposition
Consider the vector spaces $D_1$, $D_2$, $D$ and $X$ such that $D\subset X$ and $D=D_1\oplus D_2$.
Furthermore, suppose that $L:X\longrightarrow D$ is a linear map such that $D_1$ is stable under $L$...
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Prove that every two lines in space are equal or disjoint or interact at one point only
The question: Let $V$ be a Vactor space over $\mathbb{F}$, Let $\overrightarrow{v},\overrightarrow{w} \in V : \overrightarrow{v} \neq 0 $ . Then we define
$$L_{w,v}=\{\overrightarrow{w}+\...
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Understanding Equivalence of Matrix Elements in Different Bases for Hermitian Operators
Suppose $Q$ and $R$ are two system (which are represented by state vectors in the vector space V) on the same vector space $V$
$|i\rangle$ is an ortonormal base of $V$
$|i_R\rangle$ is an ortonormal ...
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Motivation behind the defination of scalar multiplication of a Vectorspace over a field
In school, we studied physical notations, such as forces, velocities, and accelerations involving both magnitude and direction. We also called any such entity involving magnitude and direction a "...
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Why is the inner product space defined separately?
While learning about the inner product space, I became curious
why it is defined separately?
In my opinion, there seems to be no difference between defining the inner product space separately and ...
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Show that polynomials with a given factor form a subspace
I have a question for one of my assignments but I don't understand how to solve it.
Let $P_n$ be the set of real polynomials of degree at most $n$, show
that
$S=\{p �� P_7:x^2+x+4 $ is a factor of $p(...
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Linear Independence without Vandermonde Determinant [closed]
Let $n > 2$ be an integer, $X_1, \ldots,X_n$ be vectors in a vector space, and $\lambda_1, \ldots, \lambda_n$ are nonzero, mutually different scalars.
I want to prove the following implication:
$$
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Two definitions of antisymmetrization of a tensor?
I am currently learning about tensors and the exterior product, and I have found some contradictory information. I have seen some sources define the antisymmetrization of a tensor as the following:
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Finding Basis for specific Spline Space
Let $S = \{s \in S: s'(a) = s'(b) = 0 \}$ be the spline space that holds all cubic splines with derivate at startpoint (a) and endpoint (b) =0. I want to find a basis for this vector space. I looked ...
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Axler Theorem 5.33: Understanding assumption WLOG
Theorem 5.33 in Axler's book is ($\mathcal{L}(V)$ denotes the set of linear map $V \to V$):
Suppose $\mathbf{F} = \mathbf{R}$ and $V$ is finite-dimensional. Suppose also $T \in \mathcal{L}(V)$ and $b,...
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Can $\text{rank} (T) + \text{nullity} (T) = \dim V$ be proven with this simple argument?
I am helping one of my friends with linear algebra and gave him this theorem to prove as an exercise:
Theorem . Let $V$ and $W$ be vector spaces over the field $F$ and let $T$ be
a linear ...
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Show that $\mathbb{R}[x]_{\leq 2}$ has a two dimensional subspace contained in the orthogonal complement of the subspace
Let $V=\mathbb{R}[x]_{\leq 2}$, and let $f$ the bilinear form given by $f(p,q)=\int_{-1}^{1} xp(x)q(x)dx.$ Find a basis $B$ of $V$ such that $[f]_B$ is diagonal, and show that $V$ has a subspace $U$ ...
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