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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3
votes
2answers
50 views

Finding a basis of a complex vector space over $\Bbb R$ given a basis over $\Bbb C$

Suppose $X$ is a vector space over $\mathbb C$ and has as basis $\{e_1,e_2,\ldots,e_n\}$. Now regard $X$ as a vector space over $\mathbb R$. What will be the basis? My thoughts: I considered ...
0
votes
1answer
20 views

Is there an error in the solution for this exercise?

I have this exercise: H is a complex hilbert space. And T is a compact operator on H. Show that if H is not separable, then 0 is an eigenvalue of T. Hint: Use lemma 1, and theorem 2. The ...
0
votes
1answer
15 views

Linear transformation of orthogonal complement

Let $T\colon \mathbb{R}^3 \to \mathbb{R}^3$ be linear. If $W$ is a $T$-invariant subspace, then $W^\perp$ (under the standard inner product on $\mathbb{R}^3$) is also $T$-invariant. Is this true? I ...
0
votes
0answers
21 views

Why do uniform, strong and weak convergence coincide for finite dimensional vector spaces?

For linear operators $A_n$, $A$ in a finite dimensional vector space $V$, I am trying to prove the equivalence of $\|A_n - A\| = \sup_{x \in \mathbb{C}^n, |x| = 1} |A_nx-Ax| \to 0$ as $n \to ...
-1
votes
1answer
24 views

Calculating normal vector to a 3d rotated plane [on hold]

I have similar question like Alex H: Calculating normal vector to a rotated plane How can i calculate normal vector to the plane that rotated around $X$ ,$Y$ and $Z$ axis (if the orginal normal was ...
2
votes
2answers
37 views

Vector in function format

Not sure how to interpret the follow: Find the intersection point(s) of the line $r(t)=(0, -2, -1)+t(1, 1, 1)$ and the plane $x+2y-4z=-3$ Does $r(t)=(0, -2, -1)+t(1, 1, 1)$ mean $r=(0, -2t, -t)$?
1
vote
1answer
18 views

Cartesian plane points given in equation form

$${x+3\over 5}={y-5\over 3}={z-11\over -3}$$ How would I determine if the above line contains the point $(1, 1, 5)$? Couldn't $(x, y, z)$ be any value? For example $(2, 8, 4)$ than the line would ...
0
votes
1answer
33 views

More on linear algebra vector subspaces

I am continuing on my journey of trying to understand vector subspaces. Question: Let $F(-\infty,\infty)$ be the set of all real-value functions defined at each x in the interval $(-\infty,\infty)$. ...
1
vote
2answers
24 views

Closed under scalar multiplication [on hold]

The subset of $\mathbb{R}^2$: $\{ (x,y)| y=\frac{7}{2}x\}$ is a subspace of $\mathbb{R}$. How can I prove that the subspace is nontrival ?
1
vote
0answers
14 views

How to Construct $N$-dimensional Unitary Matrix Basis

Galitski's Exploring Quantum Mechanics says on its page 29, (There are $N^2$ linearly) independent Hermitian matrices of rank $N$. The number of independent unitary matrices is also $N^2$, since ...
2
votes
4answers
52 views

Vector Subspace

I have a question regarding vector subspaces: show $U=\{A\in M_{22} \mid A^2=A\}$ is not a subspace of $M_{22}$. I have said: let $A={(a_1, a_2, a_3, a_4) = \begin{pmatrix} a_1 & a_2 \\ a_3 & ...
2
votes
1answer
37 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 ...
0
votes
0answers
30 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
31 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
326 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
54 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
29 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
35 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
27 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
56 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
30 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
54 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
39 views

Vector spaces and nontrivial subspace. [closed]

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
26 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
25 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
35 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? [closed]

Counting the number of sub-spaces in a finite vector space?
1
vote
2answers
29 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
69 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
30 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
103 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
52 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
30 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
54 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 ...
2
votes
3answers
63 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 ...