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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23 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 ...
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 ...
0
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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) ...
7
votes
4answers
311 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 ...
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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 ...
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2answers
878 views

Calculate Triangle Ground using Height and Top Angle

Is it possible to calculate the ground of a triangle only using the height and top angle. Click here to see a poorly draw sketch of what I'm trying to calculate. So is it possible and how, to ...
4
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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 ...
2
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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 ...
2
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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 ...
0
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1answer
95 views

inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
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 ...
2
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0answers
64 views

The canonical perspective on the Hodge star operator [on hold]

I am looking for the canonical perspective on the Hodge star operator. I want to see it done properly, not using basis for its definition, saying clearly what we assume in its definition. ...
0
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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 $$ ...
2
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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*} ...
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, ...
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 ...
3
votes
1answer
329 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
-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
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2answers
353 views

If $V$ and $W$ are subspaces of the same dimension such that $V$ meets $W^\perp$, then $W$ meets $V^\perp$

I'm having a little difficulty understanding the proof for orthogonal complements. I kind of understand orthogonal complements, but I cannot seem to find a logic to this. I'm trying to follow along ...
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2answers
55 views

Prove the Cauchy-Schwarz Inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...
1
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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 ...
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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 $\{(a, b) \mid a, b \in \mathbb{R}\}$
2
votes
4answers
55 views

Prove $\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2$

If $X,Y$ are vectors in $\mathbb{R}^n$ and $a>0$ show that: $$\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2 (*)$$ I started with ...
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 ...
4
votes
2answers
125 views

What do we call the covector associated to a vector?

Let $V$ denote an inner product space. Write $V^*$ for either the algebraic dual, or else the continuous dual. In either case, for each vector $v \in V$, we get a covector $v^c \in V^*$ given by: ...
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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 ...
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0answers
17 views
1
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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
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.
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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 ...
0
votes
1answer
39 views

Is any linear combination of arbitrary elements in a vector space also arbitrary?

Assuming $K$ is some vector space, is it valid to say the following: If $a, b, c \in K$ are arbitrary and $\gamma$ and $\phi$ are scalars, then $a+b$, $a+c$, $a+b+c$, $\gamma a$, $\gamma b$, ...
0
votes
3answers
81 views

Prove that $\mathbb{R}^∞$ is infinite-dimensional.

Prove that $\mathbb{R}^∞$ is infinite-dimensional. The section that contains this problem deals with the idea of a basis, so the proof probably has something to do with it (since a basis must ...
0
votes
2answers
36 views

Proving that the matrix of a linear transformation with respect to two bases has a particular form

I'm doing the conceptual exercises from my linear algebra book, and I ran up to the following exercise: Let $\mathbb{V}$ be a vector space with basis $\mathcal{B} = \{ \mathbf{v}_1, \ldots , ...
2
votes
3answers
70 views

Must a basis for an $n$-dimensional vector space have $n$ vectors?

Does a basis for an $n$-dimensional vector space have to have $n$ vectors? For example, if I form a basis for $\mathbb{R}^n$, do I need at least $n$ vectors in my basis set? In other words, can I ...
1
vote
1answer
26 views

Finding a vector in $\mathbb{R}^2$ given its coordinates with respect to a given basis

Consider the basis $B$ of $\mathbb{R}^2$ consisting of vectors $\begin{bmatrix}3 \\ -5 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ -5 \end{bmatrix}$. Find $x$ in $\mathbb{R}^2$ whose vector relative ...
0
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2answers
57 views

Is “basis times square matrix” a new basis?

Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis for $V$. Now we take an arbitrary square matrix $S \neq 0$. $BS$ is just a linear combination of $B$. Thus $BS$ should be a new ...
1
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1answer
22 views

Possible inconsistency of column representation with orthogonality of vectors

Let's say I have two vectors $v_{1}$ and $v_{2}$ which form a basis for $\mathbb{R}^2$. Any vector $v$ in $\mathbb{R}^2$ can be represented as $$v = av_{1} + bv_{2}$$ for some $a,b \in \mathbb{R}^2$. ...
1
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2answers
72 views

Why people use the Gram-Schmidt process instead of just chosing the standard basis

I really can't find a reason for going through all the work of the Gram-Schmidt method to make a new orthogonal basis $B'$ given an old basis $B$. If I want to change to an orthogonal basis, the most ...
1
vote
1answer
18 views

Finding a matrix representation of the linear transformation $T\colon P_2\to P_2$ ($T(f) = f''+2f'-f$)

Find a matrix representation of the linear transformation $T: P_2( \mathbb{R} ) \to P_2(\mathbb{R} )$, where $T$ is defined as $T(f(x)) = f''(x)+2f'(x) -f(x)$. I know the standard ordered basis ...
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 ...
0
votes
1answer
22 views

Is it possible to partition a basis $S$ of a Euclidean vector space into a basis for a subspace $U$ and its orthogonal complement?

Let $V$ be a Eucledean vector space with a basis $S=\{a_1,a_2,\ldots,a_n\}$. Denote $U$ be a proper subspace of $V$. Can $S$ be partitioned into the union of proper subsets $S_1,S_2$ such that ...
0
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1answer
40 views
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 ...
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
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 ...