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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0answers
22 views

Find a spanning set of the following subset of $R^3$?

Not sure how to go about this one, I'm pretty stuck: Find a spanning set of: $$\{a + c, c - b, 3c) : a,b,c \in R^3\}$$ All I know is that I need 3 vectors since the subspace is in $R^3$.
2
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3answers
62 views

Given a subset $S$, is there any universal construction (property) of $L(S)$, the linear span of $S$?

Given a subset $S$ of a vector space $V$, is there any universal construction (property) of $L(S)$, the linear span of $S$ (like the way we can construct the fraction field of an integral domain)?
0
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1answer
24 views

Prob. 10, Sec. 4.2 in Kreyszig's functional analysis book: There is a linear functional for every sublinear functional …

If $p$ is a sublinear functional on a real vector space $X$, then there exists a linear functional $\tilde{f}$ on $X$ such that $-p(-x) \leq \tilde{f}(x) \leq p(x)$ for all $x \in X$. How to prove ...
0
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1answer
16 views

Integration of vector valued functions

Suppose we have a vector space $V$ over a field $\mathbb{R}$ and $\psi:\mathbb{R}\mapsto V$ is a function. Is there any definition in general for the quantity $$\int_U\psi(x)dx$$for some $U\subset ...
0
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1answer
61 views

The nullity of a square matrix with linearly dependent rows is at least one.

The nullity of a square matrix with linearly dependent rows is at least one. True or False? Here is the answer my textbook gives: True; if the rows are linearly dependent, then the rank is at ...
4
votes
1answer
55 views

Can someone explain to a calculus student what “dual space is the space of linear functions” mean?

I ran across this phrase today in a post and I am slightly confused. From my understanding, the dual space is the space of functions that sends a vector to a real number. There are two confusions: ...
1
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2answers
54 views

Modules isomorphism

Studying vector spaces, we can find the well known result that every vector space of dimension $n$ over a field $k$ is isomorphic to $k^n$. Is there a similar theorem for modules? Thanks guys!
1
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0answers
13 views

Support of a Random Variable and Linear Subspaces in $\mathbb{R}^d$

I am reading a Text about Single Index Models where a theorem is given for the identification in case all covariates are continuous. The theorem states these four conditions: $G$ is differentiable ...
4
votes
1answer
596 views

All Invariant Subspaces of a Linear Transformation

I got this problem: Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that all it's eigenvalues are 1, 2 and 3 and the corresponding eigenvectors are $v_1, v_2$ and $v_3$ ...
0
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3answers
38 views

Is this space complete or is it incomplete?

Show (if possible) that the space of all complex sequences $x=(x_n)$ with only a finite number of terms nonzero (the number of nonzero terms may be different for different members of the space) is ...
2
votes
1answer
22 views

Denote $Z$ as the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. Find the maximum, of the number of points in $Z \cap V$.

Denote $Z$ as the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. Let $k$ be given, $0 \leq k \leq n$. Find the maximum, over all vector subspaces $V \subset \mathbb{R}^n$ of ...
1
vote
1answer
18 views

Equivalence of these propositions

Let $V$ be a $ \Bbb{K}$-vector space. Let $S$ be a subset of $V$, the $S$ is a subspace of $V$ if and only if: 1) $0_V \in S$ 2) $v, u \in S \implies v+u\in S$ 3) $c \in \Bbb{K}, v \in S \implies ...
1
vote
1answer
50 views

Vector Subspaces Proof

I'm having a hard time trying to prove the following question: I tried to use de definition of Span, but it just got redundant and got me running in circles.. Here`s what I've got so far... ...
5
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0answers
136 views

Eigenvalues of a difference of non commuting products

Let $A$ be an $n \times n$ complex matrix and let $T(M) = AM-MA$. I need to determine the eigenvalues of $T$ in terms of those of $A$. This was an exercise from Artin, and I was not being able to ...
2
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0answers
19 views

Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
1
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0answers
24 views

A question on vector components with respect to an embedding on a manifold

I've been working my way through Nakahara's book "Geometry, topology and physics" and I'm currently reading through chapter 7 (on Riemannian geometry). In the 2nd section of the chapter he gives a ...
1
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0answers
52 views

Can someone explain to me the mathematics behind this code which calculates the “minimum cosine distance”?

Suppose we have two equal sized arrays of vectors $$ array_1 = \{ (x_1, y_1), (x_2, y_2), ... , (x_n, y_n) \}\\ array_2 = \{ (x_1', y_1'), (x_2', y_2'), ... , (x_n', y_n') \} $$ What the code does is ...
4
votes
3answers
1k views

What is the main difference between a vector space and a field?

In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is ...
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votes
3answers
53 views

Proof in inner product [closed]

I am learning for my finale and having issues proving the two statements: $$ \|U\|=∥V∥ \iff\langle U+V,U-V\rangle=0 $$ $$ \|U\|^2=\|V\|^2=\langle U,V\rangle \rightarrow U=V $$ your help will be ...
6
votes
1answer
73 views

About $\mathbb{R}$ as a vector space over $\mathbb{Q}$

I want to understand better the structure of the vector space $\mathbb{R}$ over $\mathbb{Q}$. I know that it is an infinite dimensional vector space with a non countable Hamel basis, and it is cited ...
1
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2answers
63 views

Defining the $L^2$ norm of a vector valued function

I am considering a collection of function of the type, $ f:[0,2\pi]\rightarrow \mathbb{R^2}$. I want to define the $L^2$ norm of the function in that space. I am defining the a norm of ...
1
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0answers
20 views

Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$? In the case of $G = SL_2$, we have $\mathbb{C}[SL_2] = \langle a,b,c,d\rangle / (ad-bc-1 )$ and ...
0
votes
1answer
51 views

Linear Independence of Pre-Image

If $t:V \to W$ is a linear transformation from vector space V to W and A is a subset of V then it seems that if $t(A)$ is linearly independent in W then A is linearly independent in V. See for example ...
1
vote
2answers
41 views

Cauchy sequence which does not converge example.

Consider the normed space $(X, \Vert \cdot\Vert) $ where $$ X=\{ (a_n)_n \quad|\quad (a_n)_n \text{ real sequence with } \lim_{n\to \infty}a_n=0 \} $$ and $$\Vert (a_n)_n\Vert:= \sum_{n\geq ...
6
votes
1answer
53 views

Given a vector space with two inner products, there is a linear transformation taking one to another

I am looking for some hint to the following question: Let $V$ be an $n$-dimensional real inner product space and let $\langle x,y\rangle$ and $[x,y] $ both be two different inner products on V. ...
2
votes
1answer
6k views

What's the relationship between singular, nontrivial and linear dependent?

I understand that if a matrix is singular, it has no inverse. If it has nontrivial solutions, it means at least one solutions exists. If it is linearly dependent, it means that for $a_1 ...
0
votes
0answers
44 views

Equation of a hyperplane tangent to a level set

Can anyone help? The normal equation of a hyperplane is But I am unsure how you would specify that it is tangent to the point $(x_0, F(x_0))$ or to the level set.
0
votes
0answers
34 views

Linear Algebra -Subspaces

Are the following subsets of $\Bbb R ^n$ subspaces? Justify the answer. a) The set of all vectors $(x_1, x_2, \dots, x_n)$ for which $x_1x_2x_3...x_n = 0$. For this one I worked it out and got that ...
1
vote
2answers
976 views

Distance between two parallel Hyperplanes

Currently studying hyperplanes, and trying to understand the derivation on these slides: http://webdoc.nyumc.org/nyumc/files/chibi/user-content/Final.pdf check page 36. The third equation is: ...
-1
votes
1answer
45 views

Find the equation of a budget hyperplane in R4, from an endowment point and a price vector [closed]

http://imgur.com/ofDS6MO Its been a while since I had to deal with vectors, if someone could help me along with 1.1 that would be very much appreciated
5
votes
1answer
200 views

General Steinitz exchange lemma

Where can I find a proof of the following general Steinitz exchange lemma: Let $B$ be a basis of a vector space $V$, and $L\subset V$ be linearly independent. Then there is an injection ...
1
vote
1answer
44 views

Find $L\left( \left[\begin{matrix} 3 \\ -1\end{matrix} \right]\right)$?

I have the following linear map: $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Now, suppose $L\left( \left[\begin{matrix} 1 \\ 1\end{matrix} \right]\right) = \left[\begin{matrix} 1 \\ 4\end{matrix} ...
0
votes
1answer
36 views

Cross product of the gradient of two functions

I am having a bit of a confusion with some claims I keep finding on a book of Fluid Dynamics. Let's say we have two functions in 3-D space, $f(\mathbf{x})$ and $g(\mathbf{x})$, with the following ...
2
votes
1answer
54 views

Understanding Norms on Vector Spaces

Let $\|\cdot\|$ be a norm (not necessarily the standard norm) on $\mathbf R^2$ and $S$ be the set of all the vectors $v$ such that $\|v\|=1$. For any point $p\in S$, let $\ell_p$ denote the line ...
2
votes
1answer
44 views

What is the tensor product with zero factors?

I am trying to understand what tensor products are, but one particular thing confuses me slightly. For $V$ a vector space over a field $\mathbb{k}$, why do we have that $V \otimes \mathbb{k} \cong V$? ...
2
votes
5answers
216 views

If $n=\dim(V)$ and $n$ vectors are linearly independent, then they form a basis

If $V$ is a vector space and $v_1, v_2, . . . , v_n \in V$ span $V$, and $u_1, u_2, . . . , u_m ∈ V$ are linearly independent, then $m\le n$. Use this to prove that if $V$ has dimension $n$ and $u_1, ...
0
votes
1answer
32 views

Find a vector equation $r(t)$ for the line through point $(-9,-2,5)$ which is normal to the surface at $(-9,-2,5)$

Can someone help me on this? I don't know which equations to use. I don't understand what they mean by "surface".
1
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1answer
38 views

how to understand this identity about the range in linear algebra

I see the identity in page 48 of paper http://arxiv.org/pdf/0909.4061.pdf. Specially, if $U^TU=UU^T=I$, then we will have $U^T\text{range}(M)=\text{range}(U^TM)$, where $\text{range}(M)$ means the ...
0
votes
1answer
18 views

Why the following $dim(N(A)) + dim(C(A^T)) = n$ is true?

I was studying about row, column, null and left null spaces, and one thing I don't understand is why the dimension of the null space of a matrix $+$ the dimension of a the row space is equal to the ...
0
votes
0answers
18 views

On bilinear forms: if $B$ is non-degenerate, then $(U_1\cap U_2)^{\perp_L} = U_1^{\perp_L} + U_2^{\perp_L}$

I think I have a proof, but I'm not completely certain it is correct: Because $B$ is non-degenerate, it follows that if $W \leq V$, then $W = W^{\perp_L \perp_R}$. It is evident, too, that $(U_1 + ...
4
votes
3answers
2k views

Equation for non-orthogonal projection of a point onto two vectors representing the isometric axis?

Suppose I have two vectors that are not orthogonal (let's say, an isometric grid) representing the new axis. Suppose I want to project a point onto these two vectors, how would I do it? Dot product ...
0
votes
0answers
17 views

Projecting/Mapping 2D vector graphics onto 3D-models (like STL-files)

At my working place, I have the task to map or project 2D-vector graphics onto 3D-models (like stickers). This is not at all about rendering, but I would need the exact 3-dimensional coordinates of ...
2
votes
1answer
13 views

Dimension of set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$

Let, $V$ be a set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$. Then dimension of $V_F$ is (A) $\left(\begin{matrix}n\\d\end{matrix}\right)$ (B) ...
1
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0answers
54 views

Change of basis formula proof

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let $M$ be an $n$-dimensional manifold and let $(U,\phi)$ and $(V,\psi)$ be two overlapping ...
0
votes
1answer
27 views

How to find normal vector to a given set of vectors?

Given a vector space $V$ with dimension $n$, if I am given $m<n$ linearly independent vectors $a_1,a_2...,a_m$, will there always be a vector $v \in V$ which orthogonal/normal to all the given $m$ ...
1
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1answer
22 views

Dual Space of an Euclidian Space is also Euclidic with a specific bilinear form

Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form, and let $\overline{\gamma}$ be defined by: $$\overline{\gamma}: V^* \times V^* \to K, \gamma(x, y) = ...
2
votes
1answer
37 views

Prob. 14, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: A Hermitian positive semi-definite form

Let $X$ be a complex vector space, and let the map $h \colon X \times X \to \mathbb{C}$ satisfy the following conditions: For all $x, y, z \in X$ and $\alpha \in \mathbb{C}$, (i) $h(x+y, z) = ...
0
votes
2answers
380 views

proof of the full exchange lemma

Let V be spanned by $\{v_1,...,v_k\}$ and let $\{u_1,...,u_k\}$ be a linearly independent subset of V, then: 1) $k\leq n$ 2) $\exists$ a spanning set $\{w_1,...,w_n\}$ for V where $w_i = u_i$ for $1 ...
3
votes
1answer
33 views

Linear combination of matrices over finite and infinite fields

Let $F\subset K$ be the fields. Let $A_1,\ldots, A_m$ be the $n\times n$ matrices over the field $F$, and $c_1,\ldots,c_m\in K$ such that $c_1A_1+\cdots+c_mA_m$ is invertible. How to prove that for ...
0
votes
1answer
26 views

determining a linear isomorphism so that two quadratic forms become equivalent

Consider the matrix $$ G = \begin{pmatrix} 3 & 1 & -2 \\ 1 & 2 & 0 \\ -2 & 0 & -3 \\ \end{pmatrix}$$ and the quadratic form $q: \mathbb{R}^3 \to \mathbb{R}$, given by $q(v) ...