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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What does $T:V\to W$ mean in vector spaces?

What does the sign $\to $ mean in contexts like: "show $T:V\to W$ is an isomorphism" or "if $T:V\to W$ is a linear transformation"...
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Proving norm on a vector space

Let C[0,1] be the set of all continuous functions f: [0,1] -> R, Prove that ||f|| = max |f(x)| ,x in [0,1], is a norm of this vector space. In a previous exercise, I already proved that C[0,1] was a ...
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1answer
39 views

Given $\sum_{i=1}^{n} \alpha_i f_i=0_E$ prove that $\alpha_1 = 0$

We have $E$ a vector space of functions $\mathbb{R} \rightarrow \mathbb{R}$ Let $a_1 > a_2 > ... > a_n$ be such that $n \geq 1$ . Let $f_1,..,f_n$ be vectors of E such that $\forall x \ \in ...
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1answer
20 views

Prove $(F+G)+H=F+(G+H)$

I'm wondering it appears simple but how could we prove : Let $F,G,H$ be three subvector spaces of $E$, prove that $(F+G)+H=F+(G+H)$ Thank you
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1answer
48 views

When can vectors of one basis be expressed as linear combination of vectors of another basis with unitary matrix coefficients?

If I have two normalized basis $\{v\}$ and $\{w\}$ for the same hilbert space of dimension $n$ ( not necessarily orthogonal ), then when can we write the following $$v_i=\sum c_{ij}w_j.....(1)$$ such ...
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1answer
40 views

How to prove $\operatorname{Span}(\operatorname{Span}(S)) = \operatorname{Span}(S)$

Given a Subset $S$ (not necessarily a subspace) of a vector space $V$, $\operatorname{Span}(S)$ indicate the smallest subspace containing $S$. I need a hint to solve the problem ...
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1answer
22 views

Differentiation of residual sum-of-squares

In a Book(The Elements of statistical learning), I see the below equation $(2)$ is derived from $(1)$ by differentiation. $$\begin{align} RSS(\beta) & = (y-X\beta)^T(y-X\beta) & (1)\\ ...
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5answers
123 views

Proving rank of $AB$ is at most equal to rank of $B$

$A=m\times n$ matrix. $B = n\times p$ matrix. Prove that the rank of of the product $AB$ is at most equal to the rank of $B$. Current state of my work: (1) First idea: show that the kernel of $B$, ...
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0answers
41 views

Action of matrix on symmetric products

Suppose that $M : V \to V$ is a linear map of a finite-dimensional vector space. This induces a linear map $M_n : \operatorname{Sym}^n(V) \to \operatorname{Sym}^n(V)$ for any $n \geq 1$. Is there a ...
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1answer
11 views

finding the vector function

Let R~ be a vector function such that R(0) = <0, 5, 3>, R′(0) = <1, 0, −2>, and R′′(0) = <3, 4, −8> Find: i. a vector equation of the tangent line to the graph of R at t = 0 now i know ...
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Prove vector space identities

Let V be a vector space, and let $x,y,z\in V$. Prove that a) $x-(y-z)=x-y+z$ b) $0x=0$ c) $(-1)x=0-x$ I think I'm making these more complex than they need to be, but could someone show me the ...
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1answer
19 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
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0answers
14 views

vector-valued function of the given curve from two given points

So I have to fin a vector-valued function of the portion of the parabola $z = 4y^2$ on the yz-plane from the point $(0, −1, 4)$ to $(0, 2, 16)$ I don't even know where to start from this, if i get ...
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0answers
31 views

Show that there is a vector $w$ in ${\rm ker}\ (T)$ such that $v=u+w$

Suppose $U$ and $V$ are vector spaces such that $T:U\rightarrow V$ is a linear map. Suppose also that $u$ and $v$ are vectors in $V$ such that $f(u)=f(v)$. Show that there is a vector $w \in \rm ...
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0answers
25 views

Basis of orthogonal complement subspace [duplicate]

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
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1answer
25 views

Proofs involving orthonormal basis

Suppose that $V$ is an inner product space. (a) Show that if $\{e_1, . . . , e_n\}$ is an orthonormal basis for $V$ , then $$||v||^2=\sum_{i=1}^{n}|\langle v|e_i\rangle|^2\quad \quad \text{for every ...
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1answer
41 views

Spans of Orthogonal complements

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
1
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1answer
22 views

Is the set $V$ = { $([t], [g], [t], [j]): t,g,j∈$Z$,[2t+j] = [0]$} a subspace of vector space $(\mathbb Z_3)^4$?

Is the set $V$ = { $([t], [g], [t], [j]): t,g,j∈Z,[2t+j] = [0]$} a subspace of vector space $(\mathbb Z_3)^4$? I am inclined to think that it is a subspace. However, I cannot find any basis for the ...
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0answers
18 views

Rotation Matrix in domain and co-domain basis

I was asked t o derive the rotation matrix counterclockwise with given angle in different domain and co-domain basis. Using what we know from trigonometry I derived the Rotation matrix as: R(Q) = ...
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0answers
27 views

Skew-symmetric non-degenerate bilinear form

If we do symplectic linear algebra on a finite-dimensional vector space $V$, then what does $$\omega(v,w) \neq 0$$ or $$\omega(v,w) = 0$$ actually tell us about the vectors $v,w$? ($\omega$ is the ...
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1answer
24 views

Planes And Lines

Given :Point $A(1,2,4)$ and plane $P: x-y+z+2=0$ How to find coordinates of point $A'$ the symmetric of point $A$ with respect to plane $P$
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0answers
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An expression of covectors acting on vectors on the tangent space of a manifold

Let $M$ be a smooth manifold. Take $p\in M$ and $(U,\varphi)$, $\varphi:U\rightarrow \mathbb{R^n}$, a chart around $p$. Let $\mathbb{R}^n\left[\frac{\partial}{\partial x_i}\right]$ and ...
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1answer
193 views

Limit of the projection of a matrix when the projection is not continuous

Consider two real matrices: the $n\times n$ matrix $A$ the $n\times m$ matrix $B$ of rank $m$, with $m<n$. Let, for $a\in\mathbb{R}$, $$S_a=A-aI_n,$$ and denote by $P_a$ the orthogonal ...
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1answer
42 views

How to find the dimension of the given vector space

Let $L=\{p(B)|\ p\ \text{is a polynomial with real coefficients}\},$ where $B =\begin{pmatrix} 0 & 1 &0\\0 & 0&1\\ 1&0&0\end{pmatrix}.$ Then the dimension $\;d\;$ of the vector ...
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1answer
9 views

Considering a basis from two different space.

For a vector space V there is a orthonormal basis. If we watch these basis from a subspace of this vector space, are they still orthonormal?
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2answers
47 views

Infinite subspaces for a vector space that cannot be spanned by a single element

If a vector space (over an infinite field) cannot be spanned solely by a single element, does it mean it has infinite subspaces? I couldn't find an example that contradicts this
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1answer
22 views

Find parametric equations for the line through the point $(0,1,2)$ that is perpendicular to the line $x=1+t, y=1-t,z=2t$ and intersects this line.

My work so far: Since the lines are perpendicular, the dot product of their direction vectors should be $0$, so $<1,-1,2>\cdot <x,y,z>=0$. But I'm not sure where to go from here. I don't ...
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3answers
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Then prove: $\vec{v} = \vec{0}$ if $\langle u,v\rangle = 0$

If $\vec{v} \in V$ such that $\langle u,v\rangle = 0$, $\forall \vec{u} \in V$. Then prove: $\vec{v} = \vec{0}$ I tired to solve by assuming that they are $\langle u,v\rangle \neq 0$ $\rightarrow$ ...
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2answers
54 views

What's the easiest way to find all $\alpha\in\mathbb{R}$ such that $\tiny\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$ is positive definite?

For which $\alpha\in\mathbb{R}$ is $$C:=\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$$ positive definite, positive semidefinite or indefinite? It seems to be a simple task, but for ...
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1answer
33 views

Find a plane that passes through the line $x-1=\frac{y-3}{-2}=z$ and is perpendicular to the plane $x+y-2z=1$

I'm mostly having trouble with the first part. How do I make sure the plane passes through the given line, $x-1=\frac{y-3}{-2}=z$? The second part seems easy enough; just set the dot product of the ...
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1answer
24 views

finite dimensional vector spaces of functions left invariant by translation

Let $E$ be a finite dimensional vector space of functions $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall f \in E, \forall t \in \mathbb{R}, x \mapsto f(x-t) \in E$. Example of such spaces ...
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2answers
41 views

Prove $ |\langle u,v\rangle| = \lVert u \rVert \cdot \lVert v \rVert$

If V is the finite dimensional inner product space, then prove the following: If $u, v \in V$ are linearly dependent, then $ |\langle u,v\rangle| = \lVert u \rVert \cdot \lVert v \rVert$ Thanks.
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0answers
25 views

How to extended a unitary operator to a larger space?

Problem (the following is the exercise problem from Neilson and Chuang) Suppose $V$ is a Hilbert space with a subspace $W$. Suppose $U: W\rightarrow V$ is a linear operator which preserves inner ...
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0answers
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Need some hints to solve a problem from “Revue de la filière mathématique”

Let $A\in M_{n}(\Bbb R)$ and $B\in M_{n,m}(\Bbb R)$ and $C=\int_{0}^{1}\exp\left(sA\right)BB^T\exp\left(sA^T\right)\,{\rm d}s$. Prove that $C$ is invertible if and only if $\sum_{i=0}^{n-1} ...
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3answers
27 views

Trying to figure out formula for deciding how to write Linear Transformation as a matrix relative to a basis

In these lecture notes: http://www.math.rice.edu/~hassett/teaching/221fall05/linalg5.pdf the formula (last line on first page) for finding a matrix relative to bases $B'$ and $B$ is: (1) $$ C_{B'}T ...
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3answers
38 views

Line of intersection of two planes

So, this question is more like two mini-questions that are subsets of a single regular-sized question. Say I have two planes: $x-z=1$ and $y+2z=3$. I'm trying to find their line of intersection. a. ...
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3answers
34 views

Help understanding Vector Space Axioms

I am having a difficulty trying to understand an axiom regarding vector spaces. There exists an element $0$ in $V$ such that $x + 0 = x$ for each $x\in \mathbb{R}$ Two examples, that I don't ...
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1answer
20 views

How to find representation of polynomial w.r.t different basis

Let $B$ be the basis of the vector space of polynomials of degree less than or equal to 2. $B = \{1, t-1,(t-1)^2\}$. Let $u = 2t^2-5t+6$. How do you find $u_b$, the coordinate vector of $u$ relative ...
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2answers
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Obout the poset of subspaces of the vector space $\mathbb{R}$ over $\mathbb{Q}$.

Let $L$ the set of all subspaces of the vector space $\mathbb{R}$ over $\mathbb{Q}$, ordered by the set strict inclusion: $V_1<V_2$ iff $\{x\in V_1 \Rightarrow x \in V_2$ and there exists $y \in ...
1
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1answer
52 views

A problem of field in abstract algebra

If $V$ is a finite-dimensional vector space over the field $K$, and if $F$ is a subfield of $K$ such that $[K:F]$ is finite, show that $V$ is a finite-dimensional vector space over $F$ and that ...
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1answer
35 views

every finitely generated vector space has a basis. Question about the proof

Let $V$ be a finitely generated vector space over a field $K$. Then $V$ has a basis. I have a question about the proof we had in lecture. Proof: $V$ is finitely generated, this means for ...
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1answer
20 views

Scalar product with parameters

How do I get the values of the parameters in this equation? $\langle x,y\rangle = x_1y_1-2x_1y_2+ax_2y_1+bx_2y_2$ I do know that this equation shows a scalar product in $\mathbb{R}^2$, but how do I ...
3
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1answer
65 views

Relationships between affine closures and convex closures

Let $V$ denote a vector space. Then the following concepts make sense: affine subset of $V$ affine closure (affine "hull") of a subset of $V$ Suppose $V$ is in fact a real vector space. Then the ...
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1answer
19 views

Proving subset of vector space is closed under scalar multiplication

Let $V$ be the vector space of all continuous functions $f$ defined on $[0,1]$. Let $S$ be a subset of these functions such that $\int_0^1 f(x) = \int_0^1x f(x)$. To prove it is closed under scalar ...
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1answer
31 views

Prove $\dim(A) + \dim(B) = \dim(A+B)$ iff $A \cap B = \{0\}$ [duplicate]

$A,B$ are subspace of a finite-dimensional vector space $V$. Show that $\dim(A) + \dim(B) = \dim(A+B)$ if and only if $A \cap B = \{0\}$. It (kind of) seems intuitive but I'm having a hard time ...
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1answer
29 views

Find transformation matrix $T$ relative to new bases

T is a linear transformation represented as $\left(\begin{array}{ccc}1 & 1 & 0 \\0 & 2 & 0 \\3 & 1 & 0 \\0 & 1 & 1\end{array}\right)$ w.r.t the standard basis. Now ...
1
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1answer
38 views

Distance between function and subspace

Let $f(x)=cos^{n+1}(x)$, where $n \in \mathbb{N}$. In the real vector space $C([-\pi,\pi],\mathbb{R})$, we consider the inner product $\int_{\pi}^{\pi} \! f(x) g(x) dx$. My question is: What is ...
1
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1answer
56 views

About subspaces of $\mathbb{R}$ as vector space over $\mathbb{Q}$.

In many texts is noted the analogy between the transcendence degree of a field extension and the dimension of a vector space, so I'm tempting to use such analogy to better understand the structure of ...
0
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1answer
30 views

What do $F(-∞, ∞)$ and $C(-∞, ∞)$ stand for?

What do $F(-∞, ∞)$ and $C(-∞, ∞)$ stand for? They are vector spaces, with $C(-∞, ∞)$ being a subspace of $F(-∞, ∞)$. $C^1(-\infty, \infty)$ is a subspace of $C(-∞, ∞)$ and is defined as the set of ...
0
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1answer
25 views

Find basis for kernel and matrix representation

Problem 4 from https://math.berkeley.edu/~ogus/Math_54-07/Exams/midsol1.pdf $\beta$ is a basis of $P_3$, the set of all polynomials of at most degree 3.$\beta = (x^0,x^1,x^2,x^3)$. Let $T$ be a ...