For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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2
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1answer
26 views

Two conflicting answers: Problem in linear algebra involving quotient spaces and T-invariant subspaces

I was presented this scary looking problem in my linear algebra class involving quotient spaces: I am given finite dimensional vector space V over the complex numbers C and linear operator $ T:V ...
1
vote
0answers
19 views

Uncountable Kronecker Delta?

If V and W are vector spaces of uncountably infinite dimension, they still have bases (according to axiom of choice). Let basis sets be $\{v_x\}_{x \in X}$ and $\{w_y\}_{y \in Y}$, and define a set ...
0
votes
1answer
20 views

Algebraic subspaces

How do I prove that $U=\{(x,y,z)|x\text{ is an integer}\}$ is not a subspace of $\mathbb{R}^3$? I understand that I have to show $U$ is closed or not closed under vector addition and scalar ...
7
votes
4answers
308 views

How can I intuitively interpret this vector operation?

In reading through some very old source code that I inherited and came across a three-dimensional Euclidean vector operation that I can't seem to gain an intuition for. Transcribing the program code ...
4
votes
2answers
49 views

$C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.

Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid ...
0
votes
1answer
24 views

Use Riesz theorem to show functional bounded

I have the linear functional: $ F(v) = \int_\Gamma v \mathbf{g}\cdot\mathbf{n} d\Gamma$ where $\Gamma$ is a (smooth) part of the boundary of a domain $\Omega$, $\mathbf{g}$ is given (assumed smooth) ...
2
votes
1answer
33 views

Can we state the triangle inequality as $|\int_D f(x) dx| \leq \int_D |f(x)| dx$

$|\int_D f(x) dx| \leq \int_D |f(x)| dx$ is just the infinitestimal version of the triangle inequality commonly presented in any book on vector spaces Can we replace the definition of triangle ...
2
votes
0answers
25 views

Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
4
votes
2answers
35 views

Parallelogram law in $L_1$ space

Exercise 5.5 from Capinski's and Kopp's book "Measure, Integral and Probability" asks to show that it is impossible to define an inner product on the space $L^1([0,1])$. In order to get this result we ...
0
votes
0answers
12 views

Projection over product of vectorial spaces

Let $(E = E_1 \times E_2 \times E_n, \|\cdot\|_n)$ be a normed vectorial space product. Can we define the orthogonal projection over $E$ as following: Let $v = (v_1, \ldots, v_n) \in E,$ then $$ ...
4
votes
2answers
54 views

Which $n$-forms are pullbacks of top forms on $\Bbb R^n$

Let $V$ be a finite-dimensional vector space. I write $F_n(V)$ for the $n$th exterior power of the dual vector space. Which elements of $F_n(V)$ can be pulled back from a top form along a linear ...
2
votes
1answer
25 views

Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
1
vote
1answer
28 views

How do I show that this topology on this linearly-ordered set is regular?

Given some linear ordered set $X$, we define a topology by the basis: all sets of the form $(a,b)$ or $(a,\infty)$ or $(-\infty,b)$, where $a,b \in X$. I need to prove that this topology is regular, ...
-3
votes
1answer
52 views

Verify the following assertion: [on hold]

Suppose that $U=\{(x,x,y,y)\in F^4:x,y\in F\}$ and $W=\{(x,x,x,y)\in F^4:x,y\in F\}$.Then $$U+W=\{(x,x,y,z)\in F^4:x,y,z\in F\}.$$ Not sure how to add these subsets. Please provide explanation.
2
votes
1answer
17 views

Determining dimension of a sum of subspaces in terms of a parameter

Problem: Consider the linear subspaces \begin{align*} U = \text{span} \left\{ (1,0,1,0), (1,a,0,a)\right\} \quad \text{and} \quad W = \text{span}\left\{(-1, a, a^2, 0), (0,1,0,-1)\right\} \end{align*} ...
-3
votes
0answers
32 views

Vector spaces and nontrivial subspace. [on hold]

Give an example of a subset of $\mathbb{R}^2$ that is a nontrivial subspace of $\mathbb{R}^2$? $\mathbb{R}^2$ as $\{(a, b) \mid a, b \in \mathbb{R}\}$
1
vote
1answer
25 views

Show that there exists a non-negative integer $r$ s.t. $ker(T^r) = ker(T^{r+1})$.

Question: Let $V$ be an $n$-dimensional complex vector space, let $T: V \to V$ be a linear transformation. Show that there exists a non-negative integer $r$ s.t. $ker(T^r) = ker(T^{r+1})$. My ...
0
votes
1answer
23 views

The spectrum of a polynomial of an operator, question about proof, why are the operators invertible?

I have a question about a proof. In the proof $\sigma(T)$ is $\{\lambda \in\mathbb{C}: T-\lambda I\text{ is not invertible}\}$. In the proof they use this lemma: Here is the proof, my problem is ...
0
votes
2answers
33 views

Distributive property of scalar multiplication over scalar addition

I need help with a simple proof for the distributive property of scalar multiplication over scalar addition. Help with proving this definition: $(r + s) X = rX + rY$ I have to prove the truth of the ...
-1
votes
0answers
17 views

Element of one subspace not part of the other? [on hold]

Counting the number of sub-spaces in a finite vector space?
1
vote
2answers
25 views

Finding orthonormal basis for a subspace $W$ of the Euclidean space $\mathbb{R}^3$.

Problem: Let $\mathbb{R}^3$ be an Euclidean space. Find an orthonormal basis for the subspace $W$ defined as $x + 2y-z = 0$. Attempt at solution: So this is a plane in $\mathbb{R}^3$, so I guess I ...
0
votes
1answer
23 views

range and kernel of linear transformation over infinite dimensional vector spaces

How to find the range and kernel of such linear transformations ? I have already gone over the literature and have found some useful helps at example 1 and example 2. However they deal with finite ...
1
vote
1answer
68 views

Should I use set notation or list notation when writing out a basis of vectors?

I think in Sheldon Axler's Linear Algebra Done Right, he makes a comment about why the technically correct way is to write vectors in lists, such as $(v_1, ... v_n)$, while many books use set ...
1
vote
1answer
25 views

Where did I go wrong with the Gram-Schmidt orthogonalisation process?

Problem: Let $\alpha = \left\{(1,2,0), (1,0,1), (2,3,1)\right\}$ be a basis vor $\mathbb{R}^3$. Apply the Gram-Schmidt orthogonalisation process to turn $\alpha$ into an orthonormal basis for ...
1
vote
1answer
28 views

What is Fourier transform of space variable? on the similar grounds what is the Laplace transform of the same?

I understand that the transform of time domain is frequency domain and the transformation of time to frequency domain is done by Fourier/Laplace transforms. I am confused about the transformation of ...
3
votes
1answer
99 views

Subring of $M_7(\mathbb{Z}_2)$ isomorphic to $\mathbb{F}_{128}$?

Let $A \subset M_7(\mathbb{Z}_2)$ be a subring such that no proper nonzero subgroup $V \subset \mathbb{Z}_2^7$ is invariant under all matrices in $A$. I suspect that $A \cong \mathbb{F}_{128}$, but ...
1
vote
1answer
50 views

Proving the Non-existence of an Orthogonal Vector in $\mathbb{R}^n$

If $X$ is vector in $\mathbb{R}^n$ with all components > 0 then is it true that a non-zero vector, $Y$, with all components ≥ 0, can not be orthogonal to $X$ ? Considering the angles that $X$ makes ...
0
votes
1answer
53 views

$Hom(V,W)$ remains unchanged when norms of $V$ and $W$ are replaced with equivalent norms.

I was thinking about the following question from section 3.4 of Loomis and Sternberg's Advanced Calculus The fact that $Hom(V,W)$ is unchanged when norms are replaced by equivalent norms can be ...
3
votes
2answers
45 views

Prove that the linear transformations are the same.

I have this lemma: If X is a complex inner product space and $S,T \in B(X)$ are such that $(Sz,z)=(Tz,z)\forall z \in X$, then $S=T$. $B(x)$ is the set of bounded linear operators from X to X. ...
0
votes
2answers
18 views

Real valued continuous functions on [a,b] form a vector space with respect to usual addition and multiplication by scalars.

Real valued continuous functions on $[a,b]$ form a vector space with respect to usual addition and multiplication by scalars. Please help to show a proof. I think it would be a laborious one. ...
3
votes
2answers
87 views

Prove that $V = \ker T \oplus \text{Im}T$

Let $T:V\to V$ such that $f_T = \sum_{i=0}^n c_ix^i$ and $c_1 = c_n = 1, c_0 = 0$. Prove that $V = \ker T \oplus \text{Im}T$. My thoughts so far: For some basis $B$, we have $[T]_B = A$. We know ...
0
votes
2answers
28 views

What can I say about the constant of a Lipschitz condition for a scaled norm?

Let's say $X$ is a vector space with inner product $\langle \cdot,\cdot\rangle$ and induced norm $\|\cdot\|$. Then for a scalar $\theta > 0$ we define $\langle \cdot,\cdot\rangle_{\theta} := ...
0
votes
1answer
13 views

Set of Riesz homomorphisms

In a text I am using it states the following: "The set of all Riesz homomorphisms between two Riesz spaces does not ordinarily have a simple structure of its own. Consider for example, the set of ...
-1
votes
1answer
27 views

Proving that a basis of an $n$-dimensional linear space must have $n$ linearly independent vectors

Okay, I understand that a property of the basis is that a $n$-dimensional linear space has to have $n$ linearly independent vectors. I don't know how to write a proof for this though.
1
vote
1answer
51 views

I need help with a simple proof for the associative law of scalar multiplication of a vectors.

I need help with a simple proof for the associative law of scalar multiplication of a vectors. If $$(rs)X =r (sX)$$ Define the elements belonging to $\mathbb{R}^2$ as ...
0
votes
1answer
27 views

Why is $U$ $T$-invariant?

Let $V$ a finite dimensional vector space and two sub-spaces, $U, W$ such that $V = U \oplus W$. Let's assume $T$ is a linear operator such that $W$ is $T$-invariant. Why is it true that $U$ is also ...
1
vote
1answer
29 views

Why isn't the square root is cancelled in this formula?

$\sqrt{\sum\limits_{i=1}^M \vec{V^2_d}(d)}$ This is the formula of the Euclidean length of a vector in the vector space. The vector $V$ has a power of 2 so it is $V^2$. Why isn't the square root of ...
0
votes
1answer
25 views

Projection of a discrete subgroup of $R^n$ [duplicate]

Let $A$ be a discrete subgroup of $\Bbb R^n$ and let $V$ be a $m<n$ dimensional $\Bbb R$-subspace of $\Bbb R^n$. Is the projection of $A$ onto $V$ a discrete subgroup? I am most interested in the ...
1
vote
3answers
62 views

Vector space or vector field?

I seem to be having a problem distinguishing between a vector space (which I know to be a set of vectors over some scalar set) and a vector field. I know that in Multivariable Calculus a vector field ...
1
vote
1answer
24 views

How can we derive the projection formula in general?

The derivation of the well-known projection formula $proj_\vec{b}(\vec{a})=\frac{\vec{a}\cdot \vec{b}}{\vec{b}\cdot \vec{b}}\vec{b}$ uses an argument based completely on geometry. We assume vectors ...
7
votes
3answers
1k views

Is there always a point with no gravitational acceleration?

Assuming that there are no 'point particles' but rather particles have finite size and density, and that the force of gravity is defined simply by Newton's law of gravitation: $$F_g = ...
3
votes
1answer
37 views

How is this the Open Mapping Theorem?

My book has this theorem which it has stated as the Open Mapping Theorem: Suppose X and Y are Banach spaces and $T \in B(X,Y)$ is surjective. Let: $L=\{T(x): x \in X \text{ and } \|x\|\le ...
2
votes
1answer
30 views

Are invertible linear operators of bounded linear operators also bounded?

I have this definition in my book: Definition: Let X,Y be normed linear spaces. An operator $T \in B(X,Y)$ is said to be invertible if there exists $S \in B(Y,X)$ such that $ST=I_X, TS=I_Y$, ...
0
votes
1answer
31 views

Equation of the hyperplane that passes through points on the different axes

We work over $\mathbb{R}^N$. I have a set of points, each of which is on a different axis. For instance, when $N=3$ the set is given by $S=\{ (p_1,0,0);(0,p_2,0);(0,0,p_3) \}$, where $p_1$, $p_2$, and ...
1
vote
0answers
28 views

how to test if Linear Discriminant Analysis (LDA) I implemented works?

I have implemented Linear Discriminant Analysis (LDA) in C by referring various sources. Now, I would like to test the system with a simple configuration. How can I do that? I work on a speech ...
2
votes
2answers
78 views

Proving that a space is complete

There is something that bugs me about the proof I've been shown that $C(\Omega)$ (the space of continuos function on $\Omega$, a compact subset of $\mathbb R^n$) with the $\sup$ norm is complete. ...
0
votes
2answers
31 views

Possible definition of the matrix representation of a linear transformation with respect to given bases

Let $E$, $F$ be vector spaces with basis $\{e_1,\dots,e_m\}$, $\{f_1,\dots,f_n\}$. Let $T:E\to F$ be a linear transformation. We say that the matrix $A\in\mathbb{R}^{m\times n}$ represents $T$ with ...
1
vote
0answers
44 views

Linear Algebra: Two independent vectors in a 3 space?

I am trying to learn linear algebra through Strang's book. It says that all linear combinations of two independent vectors say, $(0, 0, 1)$ and $(1, 1, 0)$ will geometrically form a plane in a $3$-d ...
2
votes
1answer
73 views

Hierarchy of Mathematical Spaces

I really got lost among all those many different spaces in mathematics, and I got really confused what is special case of what. For example, I knew for long time vector spaces, then Hilbert spaces, ...
2
votes
1answer
26 views

Geometric concept of $A$-orthogonality, $A>0$

Assume the following is in in $\mathbb{R}^n$ 1. If $d_i,d_j$ are orthogonal with $i \neq j$, it means $d_i^Td_j=0$. 2. If $d_i,d_j$ are $A$-orthogonal with $i \neq j$, it means $d_i^TAd_j=0$. In ...