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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Parameterization of a “concurrent line”

What is a valid parameterization for a general, real intersection of two surfaces: $$ f(x,y,z) = 0, \, g(x,y,z) =0 ? $$ For particular cases we eliminate a coordinate if possible and use the form ...
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3answers
47 views

Find a basis of a subspace defined by a linear equation

Let $B=\{v_1,v_2,v_3,v_4\}$ be a basis of $V$. Let $$V \supset S= \left \{v:v=\sum\limits_{i=1}^4 \alpha_iv_i, \alpha_1+2\alpha_2+\alpha_3-\alpha_4=0 \right \}$$ Find a basis of $S$. I don't ...
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How to prove there exists a unique linear map such that $T(e_i) = w_i$ in an infinite-dimensional vector space?

Problem: (a) Let $V$ and $W$ be two finite dimensional vectorspaces over a field $F$, and let $\left\{e_1, e_2, \ldots, e_n\right\}$ be a basis for $V$. Then there exists for each $w_i \in W$ an ...
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1answer
42 views

Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$

Assume we have already proved this for unital algebras. Here's my book's solution: Construct the unital algebra $A^1$ [with unit $(1,0)$] as an algebra on the vector space $k\oplus A$ with the ...
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Prove {$v_1,v_2,w$} is a basis for vector space _V_

A problem from my textbook states: Let {$v_1,v_2,v_3$} be a basis for a vector space $V$. Prove that, if $w$ is not in $sp(v_1,v_2)$, then {$v_1,v_2,w$} is also a basis for $V$. Assume ...
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4answers
60 views

If $V$ is a vector space, then, proving that…

I have a big problem with this problem... : If $V_m(\mathbb{R})$ is a vector space whose dimension is "$m$" then Proving that "$m$" is even number if and only if exist an endomorphism $J$ of ...
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4answers
92 views

Is the given subset a subspace of the given vector space?

The set of all polynomials of degree greater than 3 together with the zero polynomial in the vector space P of all polynomials with coefficients in $\Bbb R$. Let $S$ be the set of all polynomials ...
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2answers
36 views

Distance between point and plane - why use the dot product?

So according to this, the signed distance between a point and a plane will be the dot product of the plane's normal vector (does it have to be a unit vector?) and the point-in-plane minus the point ...
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5answers
169 views

Find the dimension of a vector subspace

I'm doing a problem on finding the dimension of a linear subspace, more specifically if $\:$ {$f \in \mathcal P_n(\mathbf F): f(1)=0, f'(2)=0$} is a subspace of $P_n$, what is this dimension of ...
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1answer
35 views

All surfaces through a common “concur-line” [on hold]

Find all second degree surfaces passing through a common given parameterized space curve of intersection: $$ (x,y,z) = (\pm \sqrt {2 t ( 1-t)} , t , (1-t) ) $$ using a single variable parameter ...
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1answer
17 views

What is “Real coordinate space”?

What is the Real Coordinate Space in the discussion of vectors? How does it relate to Cartesian Coordinate System and Euclidean Space? P.S. Please, use naive terms.
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8 views

Softmax Regression Gradient Derivation

I'm implementing softmax regression and am deriving the max-log-likelihood update for gradient descent by hand first. Coming from the Stanford UFLDL site, they show the gradient of the cost function ...
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1answer
54 views

Consequences of the positivity condition $v^t A v > 0$ for the eigenvalues of $A$

Let $A$ be an $n \times n$ symmetric real matrix with n distinct eigenvalues $\lambda_1 , ... , \lambda_n$. a) Suppose $v^t(Av)$>0 for all v in $R^n$, v$\ne$0. Show that all $\lambda_i$ are positive ...
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2answers
46 views

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

For example, for $\mathbb{R}^n$, if I form a basis, do I need at least $n$ vectors in my basis set? In other words, can I form a basis for $\mathbb{R}^n$ by using only $n-1$ or less number of ...
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1answer
17 views

Finding change of basis matrix when given two bases as a set of matrices

Find the change of basis matrix between the following bases: $\alpha = \left\{ \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}, \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}, ...
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1answer
31 views

Generating vectors in a non-orthogonal 3D lattice with increasing magnitude

I am trying to build an algorithm to generate a sequence of lattice vectors $\mathbf{v}_n$ in 3D such that: (a) the first vector $|\mathbf{v}_1|$ is the shortest vector of the lattice (b) for all $i ...
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3answers
18 views

Determining the formula for a linear map

Determine the formula for the following linear map: $L : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with $L(1,2) = (0,-1)$ and $L(-1,-1) = (2,1)$. Attempt at solution: On the basis of these examples I ...
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1answer
11 views

Sesquilinear Forms: Polarization

This thread is only Q&A.* Given a Hilbert space $\mathcal{H}$. Consider the transforms: $$q[\varphi]:=s(\varphi,\varphi)\quad s(\varphi,\psi):=\frac{1}{4}\sum_{\alpha=0\ldots3}i^\alpha ...
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0answers
15 views

Linear (In)dependence and other relations

(i) "nontrivial solution" same as "linear dependence" same as "determinant zero" same as "the vectors lie in the same plane". (ii) "trivial solution" same as "linear independence" same as ...
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0answers
30 views

Sesquilinear Forms: Cauchy-Schwarz

This thread is related: Parallelogram Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose it ...
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2answers
37 views

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ?

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ? Or , does every infinite dimensional normed linear space has ...
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1answer
45 views

What is the 2d equivalent of vector multiplication? [on hold]

If two three-dimensional vectors, v1 and v2, are multiplied (i.e. dot product), the result will be a 3x3 matrix. If, instead, there are two three-by-three matricies, what is the corresponding ...
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2answers
38 views

Let V be a vector space of dimension n. Prove that no set of n - 1 vectors can span V.

I'm not sure I understand the question. As far as I understand it when it says vector space of dimension n, it signifies that there will be n amount of vectors; right? So basically it wants you to ...
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4answers
33 views

Prove or disprove that the set of polynomials of degree greater than or equal to two, along with the zero polynomial is a vector space

This was disproved by giving the example: $$(x^2)+(1+x-x^2)$$ The result is NOT in the set so it's NOT closed under addiction, so NOT a vector space. But I was looking for some prove that doesn't ...
2
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1answer
32 views

Suppose $U=Span\{u_{1}, u_{2} \}$ for $u_{1}, u_{2} \in U$ and $V=Span\{ v1, v2\}$ for $v_{1},v_{2} \in V$. Prove that $U+V=Span\{u1,u2,v1,v2\}$.

This is what I have so far, I don't know if this is where I stop or if there is more to prove? $$U+V = (c_{1}u_{1} + c_{2}u_{2}) + (c_{1}v_{1} + c_{2}v_{2}) = c_{1} (u_{1}+v_{1}) + ...
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3answers
35 views

Proving $\phi: V \rightarrow \mathbb{R}^n$ is linear and finding matrix representation of it

Problem: Let $V$ be a $n$-dimensional vectorspace and let $\beta = \left\{v_1, v_2, \ldots, v_n\right\}$ be a basis for $V$. Prove that the coordinate map $\phi_{\beta} : V \rightarrow \mathbb{R}^n$ ...
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1answer
23 views

Proof to show that sums of vectors spanning a vector space also span a vector space

Let vectors $v_1, v_2, and v_3$ span a vector space $V$. Show that the vectors $v_1, v_1 + v_2$ and $v_1+ v_2 + v_3$ also span $V$. How would I go about proving this? I understand that I have to show ...
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1answer
30 views

Definition of the vector cross product

As far as I understand the cross product between two vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{3}$ is defined as a vector $\mathbf{c}=\mathbf{a}\times\mathbf{b}$ that is orthogonal to the plane ...
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3answers
80 views

How does parametrization of the intersection of two surfaces induce a space curve?

Given a two surfaces say: $z=1-y$ and $ x^2+y^2+z^2=1$, we find that they intersect at: $$x^2-2yz=0$$ How is the above a space curve? Is it not just another surface? And why do we need to introduce ...
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0answers
21 views

Counting chain maps

Let $\mathbb{K}$ be a field and let $C_{\cdot}$ and $K_{\cdot}$ be bounded chain complexes with coefficients in $\mathbb{K}$. Then the set of chain maps $f_{\cdot}:C_{\cdot}\to K_{\cdot}$ is a ...
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1answer
31 views

Determining kernel and image of linear map

Problem: Which of the following maps are linear? Determine the kernel and the image of the linear maps and check the dimension theorem. Which maps are isomorphisms? 1) $L_1: \mathbb{R} \rightarrow ...
3
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3answers
34 views

Vector Valued Functions, Find some value at point

Suppose that $r$ is a vector valued function of $t$. Now, $r_0=\langle 2,2,2\rangle$ and $r_1$ is in the $y,z$ plane. If $r' \times \langle 2,3,4\rangle=0 \forall t$, how can I find what $r_1$ is? I ...
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1answer
28 views

Proving that $V = U_1 \oplus U_2 \oplus \ldots \oplus U_k$.

Problem: Let $V$ be a vectorspace and $\beta$ a basis for $V$. Now make a partition of $\beta$ in a disjoint union of subsets $\beta_1, \ldots, \beta_k$ and let $U_i = \text{span}(\beta_i)$ for every ...
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2answers
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Prove $\exists$ $v \in V$ so that $(v , f(v))$ is a basis of $V$ [on hold]

maybe you guys can help me with this one. Let's say we have a vector space $V$ with $dim(V) = 2$ and we have a linear map $f : V \rightarrow V$ with $f^2 := f \circ f = 0$ ...
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2answers
62 views

Consider the vector space V = {(a, 1 + a) | a ∈ R} with irregular definitions of addition and multiplication

with addition and scalar multiplication defined by (a, 1 + a) ⊕ (b, 1 + b) = (a + b, 1 + a + b) k '*' (a, 1 + a) = (ka, 1 + ka), k ∈ R find a basis for V. I started off with taking the general ...
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1answer
23 views

Determine the dimension of $U+W$ and of $U \cap W$. Which sums are direct sums?

Problem: Determine the dimension of the sum $U + W$ and of the intersection $U \cap W$ of the following subspaces $U$ and $W$. Which sums are direct sums? 1) $U = \text{span}\left\{(1,1,1)\right\}$ ...
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2answers
20 views

Are the difference of two vectors orthogonal if the angle between the two vectors approaches 0? (Attempted proof)

Suppose that $\vec{a}=(x,y), \vec{a`}=(x', y'), \Delta \vec{a} = (x'-x, y'-y), \theta \rightarrow 0$ where $\theta$ is the angle between $\vec{a}$ and $\vec{a'},$ and the magnitudes are equal, $a=a'$ ...
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3answers
30 views

Show that $\ker \hat{T} = \text{ann}(\text{range } T)$

This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by ...
2
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1answer
48 views

Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space) "Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed ...
2
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1answer
39 views

Finding a basis for $V, W, V+W$ and $V \cap W$

Problem: Let \begin{align*} V = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid y+z+u = 0 \right\} \end{align*} and \begin{align*} W = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid x+y = 0, z = 2u \right\} ...
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1answer
11 views

Use of GS before projecting a vector onto a plane

I need help with the following exercise: Given the vectors $u_1 = (2,-1,2), u_2 = (1,2,1), u_3 = (-2,3,3)$, what is the projection of $u_3$ onto the plane spanned by $u_1$ and $u_2$. I'm not sure if ...
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2answers
38 views

Number of vectors over a finite field that are linearily independent to a subspace

let $S$ be a vector space over a finite field of size $q$ and let $T$ be a subspace of $S$. I am looking for a formula or an algorithm to compute the number of vectors from $S$ that are independent ...
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1answer
26 views

Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...
6
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1answer
187 views

definition of ordered vector space

An ordered vector space is the pair $(V , \leq)$ where it satisfies the following: For all $x,y,z \in V, \lambda \geq 0$, i) $x \leq y \Rightarrow x+z \leq y+z$ ii) $x \leq y \Rightarrow \lambda ...
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0answers
15 views

Why do we need to worry about convergence in $\mathbb{R^Z}$ if each $\mathbf{e}_i$ are already pairwise linearly independent?

(Note: as pointed out by some users in related questions, the $\mathbb{R^\infty}$ in the link turns out to be $\mathbb{R^Z}$) Once again, a question inspired from reading this An excerpt One ...
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3answers
34 views

How to determine a basis and the dimension for this vectorspace?

Determine a basis and the dimension for the following vectorspace: \begin{align*} W = \left\{A \in \mathbb{R}^{3 \times 3} \mid A \ \text{is a diagonal matrix and} \ \sum_{i=1}^3 A_{ii} = 0\right\} ...
1
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1answer
40 views

Checking if a matrix is in the span of other matrices

Problem: Expand the following set matrices \begin{align*} \left\{ \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}, \begin{pmatrix} 2 & 1 \\ -1 & 4 \end{pmatrix}, \begin{pmatrix} 0 & ...
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0answers
15 views

Continuous action on tensor product

Let $G$ be a profinite group and $V,W$ be $k$-vector spaces with discrete topology. Suppose $G$ acts continuously on $V$ and $W$, we extend the action of $G$ to $V \otimes_k W$ by defining on simple ...
2
votes
1answer
15 views

How to see transformations on polytopes?

I have a polytope in six dimension with extreme points $(1,0,0,0,0,0)$ $(0,1,0,0,0,0)$ $(0,0,1,0,0,0)$ $(1,1,0,1,0,0)$ $(1,0,1,0,1,0)$ $(0,1,1,0,0,1)$ $(1,1,1,1,1,1)$ $(0,0,0,0,0,0)$ Each of the ...
1
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1answer
22 views

Equivalence for direct sum of vector subspaces

I have a lot of problems proving the following statement. Let $V$ be a vector space. Let $W,K\leq V,$ where $\leq$ denotes vector subspaces. $W\bigoplus K=V$ $\iff$ $K\leq V$ and is least ...