Tagged Questions
-1
votes
3answers
53 views
A question about direct sums of subspaces…
Let $W_1$ be a subspace of a finite dimensional vector space $V$. Prove that there exists a subspace $W_2\subset V$ such that $V=W_1\oplus W_2$.
EDIT$^1$:
This may be of use here:
What ...
-2
votes
2answers
37 views
$T:V\rightarrow W$ such that $R(T) \subset W'$ is a subspace of ${\cal{L}}(V,W)$ [closed]
Let $V$ and $W$ be finite-dimensional vector spaces over $F$ and $W'\subset W$ a subspace, then the subset ${\cal{L}}(V,W)$ consisting of all linear maps $T:V\rightarrow W$ such that $R(T) \subset W'$ ...
4
votes
4answers
95 views
$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$
How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
0
votes
1answer
30 views
Codimensionality: On Cardinality of Linear Equations
How does the codimension of a subspace give the number of linear equations needed to define the subspace?
1
vote
2answers
37 views
Dimension Recovery of $S \subset P_n(F)$
How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that
$f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset?
-2
votes
4answers
100 views
Dimensionality and Subspace Existence: A Potential Outlet for Disquisition
The subset of $F^n$ consisting of all vectors $(a_1,a_2,\dots,a_n)$ such that $a_1+a_2+\cdots+a_n=0$ is a subspace of $F^n$ and its dimension is ...(?)....
Initially, my intuition said the ...
0
votes
1answer
53 views
Dimensions of Matrices Range (equalities).
I’d like to find range equalities.
Considering the following:
$$
A=B+C \\
A=B.C^T \\
A=[ B^T C^T ]^T \\
$$
I would like to find the function $f$ for each equality above.
$$
dim( R(A) ) = f( R(B) , ...
0
votes
1answer
25 views
Dimension of coefficents in a density equation
The density throughout a composite material is given by
$T(x, y, z) = Axy^2 + Bxz^3 + Cy^2z^3,$
where $x$, $y$ and $z$ are the cartesian coordinates of the position inside the material.
(a) Find the ...
0
votes
1answer
126 views
Does a basis span even linear systems?
I understood basis as a set of vector $v_{1},v_{2},...,v_{n}$ as the set whose linear combination will span the entire vector space say $ \mathbb R^{n}$ which makes perfect sense in intuitive terms. ...
2
votes
1answer
141 views
Degrees of freedom
Grateful if someone could tell me whether my rationale is correct:
If I impose $m$ constraints on $\mathbb R^n$ (where $n<\infty$), then the set has $n-m$ degrees of freedom. Hence this subspace ...
