All Questions
Tagged with vector-spaces functions
208
questions
1
vote
4
answers
68
views
Linear independence proof for set of functions
How do I prove that this subset of real valued functions $\{x, \sin{(x)}, \sin{(2x)}\}$ is linearly independent. Here is the proof suggest in the book:
Consider the relationship $c_1 . 1 + c_2 . \sin{(...
- 11
0
votes
1
answer
37
views
Expressions for directional derivative
I am reading a book called "Vector Analysis" by P.R. Ghosh and J.G.Chakravorty.
In it it is stated that-
'Consider a scalar point function $f(r)$ or $f(x,y,z)$ in the neighbourhood of the ...
- 33
1
vote
1
answer
86
views
Show: $\text{im } f \cap \text{ker } f = \{0\} \iff \text{ker } f \circ f = \text{ker } f$ for vectorspace $V$ with linear function $f:V\rightarrow V$
Let $V$ be a vectorspace with linear function $f:V\rightarrow V$
Show that:
$\text{im } f \cap \text{ker } f = \{0\} \iff \text{ker } f \circ f= \text{ker } f$
This is my current proof; I'm certain ...
- 389
0
votes
0
answers
23
views
Overview understanding of space, function, and transformation by an idea of plasticine. Need some comments for correction.
I would like give a overview idea of Space, Function, and Transformation by using plasticine as an idea.
About elementary geometry mathematic, it could be said that the basic things are basic shapes ...
- 483
0
votes
0
answers
38
views
Let $f:V \to V$ be Endomorphism of vector space $V$ and $v\in V$ such that for $n\in \mathbb{N}:f^n(v)\neq 0\text{ and } f^{n+1}(v)=0$. Linear indepen [duplicate]
Let $f: V \to V$ be an Endomorphism of the vector space $V$ and $v \in V$, such that for $n \in \mathbb{N}$:
$$f^n(v) \neq 0 \text{ and } f^{n+1}(v) = 0$$
Here $f^n(v)$ means, that the map $f$ is ...
- 1,135
1
vote
3
answers
114
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Let the Homomorphism $f$ be given by $f: \text{Pol}_3 \mathbb{R} \to \mathbb{R^2}, f(p(x)) = (p'(0), p(1))$ Find a basis for Kernel and Image of $f$.
Let the Homomorphism $f$ be given by
$$f: \text{Pol}_3 \mathbb{R} \to \mathbb{R^2}, f(p(x)) = (p'(0), p(1))$$
Find a basis for Kernel and Image of $f$.
We can write a third degree polynomial as:
$$ax^...
- 1,135
4
votes
2
answers
106
views
For an endomorphism $f:V\to V$ the set of fixed points of $f$ is defined as Fix($f$) $= \{v\in V:f(v)=v\}$ Show Fix$(f)\subseteq V$ is subspace of $V$
For an endomorphism $f: V \to V$ the set of fixed points of $f$ is defined as Fix($f$) $= \{ v \in V: f(v) = v \} $
Show that Fix($f$) $\subseteq V$ is a subspace of $V$.
To be a subspace, we must ...
- 1,135
0
votes
1
answer
251
views
What would be a basis of function vector space U = {f ∈ F(X) ∶ f(−x)−f(x) = 0 for every x ∈ X}, knowing X = {−2,−1,0,1,2}.
Note : This question is from a school assignment.
Let $X = {-2, -1, 0, 1, 2}$, and set U = {$f \in F(X) \mid f(-x) -
> f(x) = 0$ for every $x \in X$}. Consider the following functions
defined on ...
0
votes
1
answer
66
views
Does $\mathbb{R}^n\times\mathbb{R}^m$ imply $\mathbb{R}^{n+m}$ mapping?
This might be a very basic question, but does the function
$$
f:\mathbb{R}^n\times\mathbb{R}^m\to \mathbb{R}
$$
necessarily imply the existence of a function
$$
g:\mathbb{R}^{n+m}\to\mathbb{R}?
$$
If ...
- 3,435
6
votes
1
answer
231
views
Vector space of functions on an empty domain?
I know that these sorts of pathological cases are irrelevant, but I want to "practice" as it were and so want to understand the following. Hoffman and Kunze give the following example of a ...
- 1,143
2
votes
2
answers
122
views
Function Vector Space and operations defined on it
How is $(af)(x)$ different from $a(f(x))$, or how is $(f+g)(x)$ different from $f(x)+g(x)$?
Intuitively, I understand that those relations need to be equal (because of the definition of vector sum and ...
1
vote
1
answer
38
views
Restriction of quotient spaces.
Let $V_{1,2}$ be two $\mathbb{R}$-vector spaces and $U_{1,2}\subset V_{1,2}$ two linear subspaces. If $f:V_{1}\to V_{2}$ is a linear map such that $f(U_{1})\to U_{2}$, it induces a well-defined map on ...
- 928
1
vote
1
answer
119
views
On the connection between vectors, functions and matrices
A vector definitely is an element of a vector space. It has often been said that all vectors are matrices (in my opinion, even this assertion needs to have a proof if we really consider vectors in the ...
- 79
0
votes
1
answer
81
views
The factor theorem for vector spaces
The factor theorem for groups is the following:
Let $f:G\rightarrow H$ be a homomorphism of groups and let $\pi:G\rightarrow G/N$ be the natural map (mapping $g\mapsto gN$), where $N$ is a normal ...
- 93
0
votes
1
answer
95
views
How does function mapping between two spaces $f \colon \mathbb{R}^2 \xrightarrow[]{} \mathbb{R}^2$ work?
This question will probably sound silly. In the mapping $f \colon \mathbb{R}^2 \xrightarrow[]{} \mathbb{R}^2$, with the function of $f(x) = x$, how does the dimension of the input equal the dimension ...
- 13