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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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 ...
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54 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 ...
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2answers
46 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. ...
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24 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. ...
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2answers
88 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 ...
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2answers
42 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} := ...
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1answer
16 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 ...
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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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1answer
57 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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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 ...
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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 ...
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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 ...
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3answers
73 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 ...
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1answer
26 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 ...
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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 = ...
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1answer
41 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 ...
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1answer
34 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 $ST=I_X, TS=I_Y$, ...
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1answer
34 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 ...
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0answers
31 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 ...
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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. ...
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2answers
33 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 ...
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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 ...
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1answer
83 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, ...
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1answer
31 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 ...
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2answers
72 views

Kernels, linear maps and their compositions.

I am reading through a paper where the author states what seems like a trivial fact. I would like to get my head around it. Let $V_1,V_2,W$ be vector spaces. Let $\alpha:V_1\rightarrow W$ and ...
2
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1answer
25 views

How does this follow from the theorem?[normed linear space]

I have this theorem: Let X and Y be normed linear spaces and let $T:X\rightarrow Y$ be a linear transformation. The following are equivalent: a. T is uniformly continuous. b. T is ...
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2answers
58 views

What is the derivative of the inner product norm on $L^2$ space?

Let $f \in L^2(X)$ such that $f$ is generated by some arbitrary constant; that is, $f = g(a)$ with $g: \mathbb{R} \to L^2(X)$. Then what can be said about the derivative with respect to some arbitrary ...
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1answer
51 views

Proving that $T:\mathbb R^N \rightarrow \mathbb R^N$ is not surjective

Let $T:\Bbb R^n \rightarrow \Bbb R^n$ be a linear transformation and let $u_1,u_2$ different vectors in $R^n$ such that for every $v \in \Bbb R^n$ $Tu_1 \cdot v = Tu_2 \cdot v$. Prove that $T$ is ...
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1answer
24 views

Closure relative to addition and scalar multiplication

I'm working with an old Introduction to Linear Algebra book by Gillet, and in the book he claims that a subset of a vector space is guaranteed to be a subspace if it satisfies closure relative to ...
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0answers
21 views

Closest line to point after non-linear map

I have a map on a vector space $M(\vec{r})$, defined as below. All components (vectors, matrices, everything) are reals in the unit range $[0,1]$. The map $M(\vec r)$ is defined as the sum of an ...
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1answer
18 views

Combine 2 vector spaces commutatively

I have a finite dimensional vector space $V$, and I want to create a "product" space $$U = V \times V$$ similar to the Cartesian product, with elements like $(v_1, v_2)$, but I want it such that the ...
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2answers
57 views

Need example of: Algebraic sum of closed vector subspaces need not be closed

I've read somewhere that given two closed subspaces $V_1,V_2$ in topological vector space $X$, their algebraic span $V_1+V_2=\{x_1+x_2 |x_i \in V_i, i=1,2\}$ need not be closed. I always thought that ...
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2answers
135 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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71 views

Vector Space spanned by the polynomials $p_n(x) = x^n$

Let $V$ be a vector Space Spanned by the set $\mathbb B = \{ p_n(x) = x^n | x \in \mathbb R , n \in \mathbb N \}$. Is $V$ is a vector Space of all real valued continous function on $\mathbb R$ ...
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0answers
20 views

$N$-dimensional linear operator is normal, Lagrange interpolation?

Is there a way to see that an $N$-dimensional linear operator $A$ is normal if and only if $A^\dagger$ can be represented as a linear combination of $I, A, A^2, \dots, A^{N-1}$ using Lagrange ...
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1answer
43 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 $a+b$, $a+c$, $a+b+c$, $\gamma a$, $\gamma b$, ...
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68 views

Proving that $V$ is a vector space, if $a+b=ab$ and $a *b=a^b$.

I am currently studying Halmos' "Linear Algebra Problem Book" and am stuck on problem 21(4). Let $V$ be the set $\mathbb{R}_+$, and let $F$ be the set $\mathbb{R}$. Let's define the sum of two ...
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0answers
25 views

Finding transformation with respect to a basis

Let $T:R^3 \rightarrow R^3$ be a non-invertible linear transformation that's represented with respect to the base: $ B = ((1,0,1),(0,1,-1),(1,-1,0))$ By the matrix: $$[T]_B=\begin{pmatrix} 1 & 0 ...
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1answer
31 views

Set a new length for a vector?

I never encountered such action. Can someone explain this on page 47? The programmer uses a "SetLength" function on a 3-dimensional vector. Here's the statement: ...
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44 views

Example of straight line and a point in $\Bbb R^3$ such that there are infinitely many planes passing through it

Give an example of a straight line $l$ in $\mathbb R^3$, given by a system of two equations, and a point $(a,b,c)\in \mathbb R^3$ such that there are infinitely many planes in $\mathbb R^3$ passing ...
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3answers
69 views

Isomorphism of vector spaces

There is an example given in my lecture notes, which I feel a bit uncertain about $ \{ $functions$ \{1,2,...,n \} \to \mathbf F \} \cong \mathbf F^n $ by the map $ f \mapsto (f(1),f(2),...f(n))$ ...
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35 views

Given two basis, find the transformation matrix from one to another

I have these two basis of $M^R_{2x2}$: $C= (\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 \\ 1 & ...
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26 views

Showing Vector Space

Let $S = \{(x,y,z)\in R^3 : 2x-3y+5z=0\}.$ show that S is a real vector space using the standard operations on $R^3$ Having trouble showing S is closed under vector addition because of the ...
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26 views

If I want to find the dimension of the image of a linear transformation…

If I have a linear transformation $T(v)=Av$ and want to find the dimension of the range$(T)$, the following procedure is valid? Looking at the columns of $A$, if all columns are linearly independent, ...
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71 views

Finding eigenvalues of $A^{10} + A^7 + 5A$.

Problem: Let $A = \begin{pmatrix} 1 & 2 & -1 \\ 0 & 5 & -2 \\ 0 & 6 & -2 \end{pmatrix}$. 1) Compute the eigenvalues of $A^{10} + A^7 + 5A$. 2) Compute $A^{10} X$ for the ...
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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 ...
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2answers
49 views

Is this matrix diagonalizable over $\mathbb{R}$ or $\mathbb{C}$?

Problem: Let $A = \begin{pmatrix} 6 & 0 \\ -2 & 2 \end{pmatrix}$. Is this matrix diagonalizable over $\mathbb{R}$? If not, is it diagonalizable over $\mathbb{C}$? Compute the eigenvalues ...
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3answers
62 views

Find eigenvalues and eigenvectors of this matrix

Problem: Let \begin{align*} A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{pmatrix}. \end{align*} Compute all ...
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1answer
22 views

In an icosahedron subdivided n times, how can I find the coordinates of adjacent centroids?

I think it would be helpful to refer to this image when trying to follow my description: http://i.imgur.com/nRXQo3W.jpg (taken from http://experilous.com/1/blog/post/procedural-planet-generation). ...
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1answer
41 views

Vector spaces - $\mathbb{R}^n$ and $\mathbb{R}^m$

I stumbled on the following text on Wikipedia: Suppose the random column vectors X, Y live in $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, and the vector $(X, Y)$ in $\mathbb{R}^{n+m}$ has a ...