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
9 views

Domain of compostions of linear mappings

Let $T$ be a linear transformation from $\Bbb R^3$ into $\Bbb R^2$ and $S$ be a linear transformation from $\Bbb R^2$ into $\Bbb R^3$. Is the mapping $ST$ a linear transformation from $\Bbb R^3$ into ...
2
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1answer
15 views

Does a vector have to be continuous to fall within a set?

The question asks: explain why $\ f(x) = $ $\ x \over \ x^2 + 4x + 3$ is a vector in $C[0, 3]$ but not a vector in $C[-3, 0]$. I know that $f$ is not continuous on $C[-3, 0]$ at $x = -1$ and $x = 3$. ...
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0answers
17 views

A thought about transition matricies in vector spaces

I am trying to work out the relationship between transformation matricies of a vector space with different bases. I came up with an equation which does not look right, but I would like your opinion. ...
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1answer
16 views

functions and the commutative property

with regard to vector spaces of functions. How do I know if the commutative property holds for a set of functions. especially if the vector space includes an infinite set. for instance, for the ...
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2answers
42 views

Vector spaces whose elements are functions

I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. It seems that functions can be manipulated as vectors as ...
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3answers
34 views

Proof of linear independence of non-empty subsets

The question states: Show that if $S = \{v_1, v_2, \ldots , v_r\}$ is a linearly independent set of vectors, then so is every non-empty subset of $S$. I understand that if $r>n$, $S$ is ...
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3answers
41 views

For which values of a do the following vectors form a linearly independent set in R^3

I've seen this same question, but asking for linearly dependent, not linearly independent. $$ V_1= \left(a,\, \frac{-1}{2}, \,\frac{-1}{2}\right),\;\; V_2= \left(\frac{-1}{2},\, a, ...
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1answer
18 views

Find vector and parametric vector of a line

I have a line that is perpendicular to a plane. This perpendicular line is $3i-2j+6k$. I've also been given that the line passes through $A(2,3,0)$. I'm unsure on how to represent this line as a ...
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3answers
22 views

Find the vector passing through a given point which is orthogonal to a given triangle in space

I'm given this problem where I have 3 points in space $A(3, -1, 2)$, $B(-2,1,2)$ and $C(2, 0, 5)$. I need to find the vector passing through point $A$ that is perpendicular to the triangle made by ...
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2answers
51 views

An example of non euclidean inner product [on hold]

Please give me an example of non euclidean inner product.Is there any method to construct such an inner product?
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1answer
22 views

Linear Algebra-Vector Subspaces Question [on hold]

Let $U=\{f\in C^1([-1,1],\Bbb R);y'=y+1\}$. Is $U$ a subspace of $C^1([-1,1],\Bbb R)$?
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0answers
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Does $\dim (A_1\otimes A_2)=\dim(V_1\otimes V_2)$ for all affine spaces $A_{1,2}$, their vector spaces $V_{1,2}$ and the operations $\cap,+$?

Let $A_1=P_1+V_1,A_2=P_2+V_2$ be affine spaces. My teacher uses $\dim$ on affine spaces and the embedded vector spaces interchangeably, which is correct by definition for $\dim A_1=\dim V_1$, but ...
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0answers
10 views

Show that W is a Subspace of R³

can you help me: Let $u=(1,2,-3)$ and $v=(-2,3,0)$ Two Vectors in R³ and let W the subpace of R³ that consists of all the vectors shape $au+bv$, where, $a,b ∈ R $ show that W is subspace of R³ im ...
1
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1answer
21 views

Correspondence between linear maps of a vector space into itself and linear maps of the dual into itself.

I was wondering about vector spaces and their dual. Specifically, in the context of finite-dimensional vector spaces, I asked myself if it is true that there is a one-to-one correspondence between the ...
1
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1answer
26 views

Connection between algebraic multiplicity and dimension of generalized eigenspace

Assume $V$ to be a finite dimensional vector space. Define the algebraic multiplicity $am(\lambda)$of an eigenvalue $\lambda$ of a linear operator $T:V\to V$ as the maximum index of the factor ...
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2answers
28 views

How to show if 2 vectors are the same

I've been given 2 lines in different forms $L1$ is $$\frac{x-1}{4} = \frac{y-2}{3} = \frac{z-10}{5} $$ $L2$ is $$x = -7-4t$$ $$y = -4-3t$$ $$z = -5t$$ I've converted $L2$ into its Cartesian form as ...
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1answer
26 views

How many elements are there in the vector space over F of dimension 5

When $F = \mathbb Z_2$ (the two element field), how many elements are there in the vector space over $F$ of dimension $5$? Would it be $32$? Thank you so much.
4
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0answers
27 views

Is the Laplace transform a vector space isomorphism? And what space is it isomorphic to?

The laplace transform is a linear transformation, $\mathcal{L}: \mathcal{M} \rightarrow?$, where $\mathcal{M}$ is the set of exponentially bounded functions on $\mathbb{R},$since ...
2
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2answers
22 views

Basis for the vector space P2

I am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial $ ax^2 + bx + c $ with the matrix $ A = \begin{bmatrix} 1,0,0 \\ 0,1,0 \\ 0,0,1 \\ \end{bmatrix} ...
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1answer
47 views

Show that $\| u - v \|^2 = \| u - P_U(v) \|^2 + \| v - P_U(v) \|^2 $ and minimize $d(u, v)$

i) Let $\left(V, \langle\ ,\ \rangle\right)$ be an inner-product space, $v \in V$, and let $U$ be a subspace of $V$ with the orthogonal projection map $P_U$. Show that $ \| u - v \|^2 = \| u - P_U(v) ...
2
votes
1answer
36 views

calculating characteristic polynomial in $\mathbb{R}^n$

Given some hyperplane arrangement $\mathcal{A}$, we call any subset $\mathcal{B}\subseteq \mathcal{A}$ $\textit{central}$ if $$\displaystyle \bigcap_{H\in \mathcal{B}}H\neq \emptyset.$$ There is a ...
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votes
3answers
42 views

Linear Algebra, kernel [on hold]

Suppose that $W$ and $V$ are vector spaces, and that $f : V \mapsto W $ 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 ...
2
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4answers
46 views

How to test if these are a vector space and find the basis?

I have been trying to work through these linear algebra questions in my text book for hours now, but i just cant seem to figure it out. The question is: ...
1
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1answer
26 views

Proof for the necessity of conditions for a subspace

In [Axler 2015], Theorem 1.34 states that A subset $U$ of $V$ is a subspace of $V$ if and only if $U$ satisfies the following three conditions: additive identity: $0\in U$; closed ...
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0answers
20 views

Divergence of scalar field and curl of scalar field?

I have V(x.y)=(y^c,x^c) for positive c and r(x.y)=(x.y) I want to find div(V X r) and Curl (V X r). So V X r is determinant and it is scalar field. I got f(x.y)=y^(c+1)-x^(c+1) Thus, div(V x r) = ...
0
votes
1answer
17 views

Finding a vector equation for a trajectory

A shell is fired from the ground with muzzle speed of 320 ft/s and elevation angle of 60 degrees (assume $g=32 \, \mbox{ft}/\mbox{s}^2$) Find a vector equation for the shell's trajectory ...
1
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4answers
64 views

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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0answers
28 views

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 ...
1
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1answer
38 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
2
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0answers
28 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 ...
3
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1answer
38 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 ...
0
votes
1answer
19 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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0answers
8 views

System of parametric equation [on hold]

Given : point $A(1,2,3)$ and line $\ell: x=m-1 ; y=-m ; z=2m-2 $ find system of parametric equation of line passing through $A$ and perpendicular to $\ell$.
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5answers
88 views
+50

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 $B$, ...
1
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0answers
37 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 ...
0
votes
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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0answers
41 views

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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0answers
11 views

Orthocenter of triangle given by vectors in $3$-dimensional space [closed]

Given: $B=(1,0,0), D=(0,1,0), E=(0,0,1)$. Question: find the coordinates of the orthocenter $L$ around triangle $BDE$. With the formula used
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1answer
15 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
12 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
30 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
24 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 ...
0
votes
1answer
18 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 ...
0
votes
1answer
39 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
vote
1answer
21 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
12 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) = ...
1
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0answers
21 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 ...
0
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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
13 views

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 ...