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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73 views

Help me perfect out my current linear algebra knowledge

My questions are along my workings, I have attempted both the parts as much as possible as I can. Please help me on this question. My question comes as (i) Is my proof perfect? (ii) Am I correct? ...
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1answer
23 views

How come that sum of two lines in a triangle equals the third line?

I have been looking into this question and cannot understand how they came to a conclusion in their solution. we know that : $$(1) :\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 $$ ...
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1answer
134 views

For $n \ge 2$ , does every linear operator on $\mathbb R^n$ has an invariant subspace of dimension $2$ ?

Is it true that for $n \ge 2$ , every linear operator $T$ on $\mathbb R^n$ has an invariant subspace of dimension $2$ ? I know that $T$ always either have a $1$ or $2$ dimensional invariant subspace ; ...
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1answer
60 views

A question on nilpotent linear operator on finite dimensional vector space with dimension same as degree of nilpotency

Let $T$ be a nilpotent linear operator of index $n>1$ ($T^n$ is the null operator but $T^{n-1}$ is not ) on a vector space of dimension $n$ ; then how do we prove that there is no linear operator ...
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4answers
56 views

Prove that for any nonzero vectors $\bf u$ and $\bf a$ in $\Bbb R^n$, the vector $\bf a$ is orthogonal to ${\bf u} - \mathrm{proj}_{\bf a}{\bf u}$.

Prove that for any nonzero vectors $\bf u$ and $\bf a$ in $\Bbb R^n$, the vector $\bf a$ is orthogonal to ${\bf u} - \mathrm{proj}_{\bf a}{\bf u}$. I'm not sure how to start proving this. I don't ...
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1answer
34 views

Let $P_1 = (x_1, y_1)$. Describe $P = (x,y)$ in $\mathbb{R}^2$ s.t. $||P-P_1|| = 9$ by identifying conic and finding its equation

Let $P_1 = (x_1, y_1)$. Describe the set of all points $P = (x,y) \in \mathbb{R}^2$ such that $||P-P_1|| = 9$ by identifying the type of conic and finding its equation. I'm sorry, but this question ...
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1answer
34 views

Linear map - how to show this?

Assuming that I have a map $A: \mathbb{R}^2 \rightarrow \mathbb{R}$ and we have $A(-x,x) = -A(x,x)$ and $A(x+y,x) = A(x,x)+ A(y,x)$. Is this sufficient to conclude that $A( \lambda x+y ,x ) = \lambda ...
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1answer
145 views

Spans and Dot Product: Findin the linear combination

Suppose $(v_1, v_2, v_3)$ is a set of vectors mutually perpendicular. Assume that $\|v_1\|= \sqrt{27}\quad \|v_2\| = \sqrt{14}\quad \|v_3\|= \sqrt{ 4}\ $ Let $w$ be a vector in ...
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2answers
60 views

proving that for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ (verify)

for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ I wasn't sure how to go about this correctly so what I did was set $v+w$ to $v$, yielding $w = v-v = 0$, since it ...
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1answer
48 views

Knowing if spans overlap

Only the first checked squares are deemed to be correct. Why is D not correct? After all, the vectors do overlap on the same plane...
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1answer
47 views

Is $Ape_1+Aqe_2$ where A (3x3) matrix, considered as a linear combination of $e_1,e_2$

$$\alpha=-8$$ Eigenvectors: $$e_1 = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} \text{ and } e_2 = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$$ What I did : (i) $x ∈ V \implies x$ of the ...
3
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1answer
86 views

Identify irrational basis of $\mathbb{Q}$-vector space

A real sequence $\mathbf{x}=(x_k)_{k\in\mathbb{N}_0}$ is of the form $$ x_k=\alpha r_k,\quad \alpha\in\mathbb{R}\backslash\mathbb{Q},\quad r_k\in\mathbb{Q},\tag{*} $$ if and only if all of its terms ...
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1answer
36 views

Matrix equations?

I came across this problem: I have successfully found the bases of the null space but I can't seem to understand the second part. I looked around online and found nothing useful. I would appreciate ...
2
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1answer
53 views

Can we define a binary operation on $\mathbb Z$ to make it a vector space over $\mathbb Q$?

It is known that any infinite cyclic group , in particular $(\mathbb Z, +)$ , can never be a vector space . So we may ask , Can we define an operation $*$ on $\mathbb Z$ such that $(\mathbb Z , *)$ ...
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1answer
131 views

Visualize and define a vector space without dot / inner product

I'm trying to rebase my know how in linear algebra, restart from scratch to get a more formal and useful set of definitions to help me with computer programming stuff . One of the first concepts is a ...
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1answer
55 views

Linear Algebra, Spans and subspaces

Let $V= \mathbb{R^3}$ and consider the following elements of $V$: $\mathbf{u}_1 =(1,2,0)$, $\mathbf{u}_2=(3,1,0)$, $\mathbf{u}_3=(1,-1,1)$. Let $U= \langle\mathbf{u}_1,\mathbf{u}_2\rangle$ and ...
0
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1answer
127 views

Simple Vector Space Rotation

Given Data in the problem & notation convenstions We have 3 rotation vectors called $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$, magnitude of these vectors will give you angle of rotation We ...
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1answer
219 views

Calculate X Y Z from two specific degrees on a sphere

I am a programmer, don't know much about advanced math. I would need the exact formula(s) that could achieve this, so I can translate it to my programming language. I am having a headache trying to ...
7
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3answers
360 views

Vector Space definition

My book lists ten axioms that must hold for a set of objects (vectors) $V$ to be called a vector space. One of those axioms is: $$1\vec{u} = \vec{u}$$ Is there a reason why this axiom must be on the ...
3
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2answers
52 views

Sets of binary sequences

In my course on linear algebra we have recently introduced linear independent subsets of vector spaces. As an exercise I have been thinking about examples of infinite linearly independent sets and ...
2
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2answers
324 views

On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
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1answer
170 views

By given equation, finding orthogonal projection

Find the orthogonal projection of line with direction vector $u = ( 1 , 2 , 0 )$ onto the plane described by equation $-3x - 2y + 2z = -2$ i have tried to search ...
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3answers
650 views

How to determine volume of parallelepiped by 4 points

Let points $(0,0,0), (1,2,x), (-2,1,0)$ and $(1,1,3)$ be at four corners of a parallelpiped. Determine the volume of the parallelepiped by using the determinant in terms of $x$. For what value of $x$ ...
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3answers
128 views

Does there exists a vector v such that $Av\neq 0$ but $A^{2}v=0$

Let $A$ be a $4\times4$ matrix over $\mathbb C$ such that $\operatorname{rank}(A)=2$ and $A^{3}=A^{2}\neq0$. Suppose that $A$ is not diagonalizable. My question is , "Does there exists a vector $v$ ...
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1answer
26 views

Cauchy sequence of vectors when dotted with another vector gives a Cauchy sequence of scalars?

My question is related to vector spaces with an inner product defined (the space is not necessarily complete i.e. not a Hilbert Space) So imagine I have a Cauchy sequence of vectors ...
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2answers
87 views

Linear Algebra proof: $\operatorname{proj}_a(\operatorname{proj}_a(b)) = \operatorname{proj}_a(b)$

How would I show the following? $$\operatorname{proj}_a(\operatorname{proj}_a(b)) = \operatorname{proj}_a(b)$$ I subbed in the projection formula of $\dfrac{a\cdot b}{\|a\|^2}a$ but I did not get ...
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0answers
23 views

plane generated by n linearly independent n-dimensional vectors

Prove that the following statement is true. I'm not sure whether the term 'linear combination (narrow sense)' is widely used since I'm studying in Korea. According to my professor, the term ...
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2answers
62 views

Prove the Cauchy-Schwarz Inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...
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2answers
151 views

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am ...
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1answer
98 views

Association of a vector space to metric, normed and inner product spaces

There is a nice visual representation of mathematical spaces from this post: I am not quite sure how vector spaces fit into this image. I know metric space is not necessarily a vector spaces, but ...
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2answers
130 views

If $A^2 =0$ then possible rank of $A$

Let, $A$ be a non zero matrix of order $8$ with $A^2 =0.$ Then one of the possible value for rank of $A$ is (a) $5$ (b) $4$ (c) $6$ (d) $8$. Attempt : As , $A^2=0$ , so $A$ is a nilpotent ...
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1answer
37 views

Calculate projection of a line in a square

Said that we have two points (P1, P2) that form a line, and 3 points (S1,S2,S3) that form a square, how would we calculate the position X and Y of the point resulting from the intersection of the line ...
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0answers
27 views

Derivative of vectors dimension do not agree

I have two n by 1 vectors $\mathbf w,\mathbf v$ with respect to $\mathbf w$, and $\mathbf v$ is some function to $\mathbf w$. so I can get a scalar from $\mathbf w^T\mathbf v$, I want to take ...
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1answer
30 views

Which of these sums is equal to $\mathbb R^4$?

I'm given the following sets: $$U=\{(0,a,b,a-b): a,b \in \mathbb{R}\} \\ V=\{(x,y,z,w): x=y, z=w\} \\ W=\{(x,y,z,w): x=y\}$$ I'm trying to determine which of the following is equal to $\mathbb R^4$: ...
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0answers
44 views

How to show that the vertices of a convex hull are given by these specific subsets…

We work over $\mathbb{R}^N$. Let $V$ be the corners of the unit cube $[0,1]^N$, or equivalently the set of vectors whose coordinates take values $0$ or $1$. Let $d:\{0, \ldots, N\} \to \mathbb{R}_+$ ...
1
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1answer
59 views

Rank of a linear transformation T

Let, $n$ be a positive integer & let $M_n(\mathbb R)$ be the space of all $n\times n$ real matrices. If $T:M_n(\mathbb R)\to M_n(\mathbb R)$ is a linear transformation such that $T(A)=0$ , ...
2
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2answers
43 views

Dimension of matrices with entries $a_{ij} = a_{rs}$ with $i+j = r+s$.

Let $n$ be a positive integer and $H_n$ be the space of all $n \times n$ matrices $A = (a_{ij})$ with entries in $\Bbb{R}$ satisfying $a_{ij} = a_{rs}$ whenever $i+j = r+s \; (i,j,r,s = 1, 2, \ldots, ...
2
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0answers
40 views

How to understand affine space and affine transformation

As far as I know, the affine space is a space without origin point. Some others define affine space as $$A=\{\sum_{i=1}^N \alpha_i \boldsymbol{v_i|\sum_{i=1}^N}\alpha_i=1\}$$ How do we relate these ...
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1answer
36 views

Do the given vectors span $\mathbb{R}^3$?

Do the following vectors span $\mathbb{R}^3$: $$v_1 = (2, -1,3)$$ $$v_2 = (4, 1, 2)$$ $$v_3 = (8, -1, 8)$$ I use Gaussian Elimination to bring the matrix to an echelon form, with a pivot of "1" in ...
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0answers
69 views

Can a Norm be Induced by two Different Complex Inner Products?

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$. If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and ...
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2answers
44 views

Show that a unique matrix exists for the coordinate vectors in a vector space

If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space $V$, there exists a unique matrix $M$ such that for any $f\in V$, $[f]_A=M[f]_B$. My textbook uses this theorem ...
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1answer
54 views

Proofs for $n$-dimensional vector spaces $V$

Suppose $V$ is an $n$-dimensional vector space. Prove that there is at most $n$ linearly independent elements in $V$. Prove that a set of $m<n$ element in $V$ cannot span $V$. I'm not really ...
0
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1answer
49 views

Find the N versors more 'spaced' [closed]

I have to deal with a concrete problem that is: Given a 3d object I want to select N directions with N integer and N>=3 for projection that would maximize the information I gain and thus my ability to ...
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0answers
58 views

Regarding Linear Subspaces over a Finite Field… TFAE:

Let $V=\mathbb{F}^n$, for a finite field $\mathbb{F}$. Prove the equivalence of the following statements: There is a linear subspace $C$ of $V$ with the property that every vector $v$ of ...
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1answer
28 views

Continious subbundle

Let $W$ be a vector bundle with base $\Omega$ and projection $p$. A continious subbundle of $W$ is a subset $W_0$ of $W$ such that $p|W_0$ defines a vector bundle over $\Omega$. Now here ...
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0answers
26 views

dim of $\Bbb R^3 \otimes_\Bbb R \Bbb C$ when considering as a $\Bbb C$-vector space

I'm looking at Sergei Winitzki's Linear Algebra via Exterior Products, and he has a question on tensor products. Firstly we construct the real vector space $\Bbb R^3 \otimes_\Bbb R \Bbb C$ which is ...
2
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1answer
55 views

Reducing Spaces: Complement

Given a Hilbert space $\mathcal{H}$. Consider an operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad\overline{\mathcal{D}(T)}=\mathcal{H}$$ Regard a subspace: ...
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1answer
53 views

Vector spaces and direct sums

The map that was constructed in lectures is: $V,W$ subspaces of $U$. $f\colon V \oplus W \to U$ by the formula: $f((v,w))=v+w$ for $v$ in $V$, $w$ in $W$ Is it correct to generalise this to, ...
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2answers
64 views

Is union of two subspace a subspace too? [duplicate]

Assume that $W$ and $V$ are two subspace of $X$. Is their union a subspace of $X$ too? I think it is not true unless under certain conditions but I do not know what conditions...
0
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1answer
50 views

Proof vector x + ⃗y = ⃗x + ⃗z then ⃗y = ⃗z

Let ⃗x, ⃗y and ⃗z be vectors in a vector space V . Prove that if ⃗x + ⃗y = ⃗x + ⃗z then ⃗y = ⃗z. No idea how to start.