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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322 views

Suppose that a cube is inscribed in a sphere of radius one. What is the volume of the cube? my reasoning vs answer

Now my reasoning is that, s^2 + s^2 = 2^2, where s is the side of the cube, giving, s^3 = 2 sqrt 2. But the answer and explanation here is different: http://math.acadiau.ca/aumc/hints4.pdf how is the ...
0
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1answer
79 views

Eigenspace and polynomials?

My prof introduced us to eigenvectors and eigenvalues today. He then gave us the following theorem: Theorem 6.6: Let $A$ be a square matrix, let $\gamma$ be an eigenvalue of $A$ with multiplicity ...
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1answer
25 views

vector problem in parallelopiped

I want to find out absolute volume the parallelopiped I have not got that how they did with this vector notation, $h= \vec{A}\cdot \vec{n} $ The volume will be $\vec{A}\cdot (\vec{B} \times ...
3
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1answer
81 views

On the definition of the direct sum in vector spaces

We say that if $V_1 , V_2, \ldots, V_n$ are vector subspaces, the sum is direct if and only if the morphism $u$ from $V_1 \times \cdots \times V_n$ to $V_1 + \cdots + V_n$ which maps $(x_1, \ldots, ...
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1answer
625 views

Dimension of direct sum of vector spaces [duplicate]

Let $V$ and $W$ be finite dimensional vector spaces on a field $F$. Show that $\dim(V\oplus W) = \dim V +\dim W$. My idea: let $\dim V=n$ and $\dim W=n$. So $\mathcal{A}=$ {${v_1 , v_2 ,... , ...
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1answer
40 views

Eigenvector and Its Span

Let $V$ be a vector space over the field $F$ and let $T$ be a linear transformation from $V$ to $V$. Let $v\in V$ such that $v\neq 0$, let $W=span\{v\}$. Prove that if $T(W) \subset W$, then $v$ is an ...
11
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1answer
509 views

Why it is important for isomorphism between vector space and its double dual space to be natural?

I'm reading the book (by A. Kostrikin) on linear algebra and I feel like I'm really missing something about this idea. I understand the formal proofs of: a) isomorphism between vector space $V$ and ...
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1answer
28 views

Find the vector so that it is parallel

Find a vector with length 7 and is parallel to the line $y=\frac{12}{13}x-1$ Is there some more advanced mathematics behind this than elementary maths or? Could use a nudge.
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1answer
29 views

linearly independence of functions over $\mathbb{R}$

I have been asked to prove that $\{\tan(ax)|a\in \mathbb{R}^+\}$ is linearly independent. I was wondering if there is a generic method/idea for proving linear independence of functions over ...
0
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2answers
25 views

Vectors Dependent or not?

Am I doin it right? $3-2x + x^2,6-4x+2x^2 $ in $P2$ Check if they are dependent or not? I am taking their determinant as written $| 3-2x\space\space\space\space\space\space\space\space\space 6-4x ...
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1answer
57 views

Show that $W$ is NOT a subsbace of $ \mathbb{R}^3$

$W $= {$(a,b,c):a^2 + b^2 + c^2 \leq 0$} As far as I have tried it with my concepts which are somewhat definitely not cleared the W is a subspace of $\mathbb{R}^3$ but I don't know why not. I tried ...
0
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1answer
33 views

3-dimentional vector space over the field $\,F_3=\Bbb Z/3 \Bbb Z$

Let $V$ be a 3-dimensional vector space over the field $\,F_3=\Bbb Z/3 \Bbb Z$ of $3$ element.Then what is the number of distinct 2-dimentional subspace of $V$ ?
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4answers
487 views

Are all subspaces of equal dimension (of a vector space) the same?

I haven't quite gotten my head around dimension, bases, and subspaces. It seems intuitively true, but are all subspaces of equal dimension of the same vector space the same? If so, does it follow ...
0
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1answer
34 views

Expressing a vector as the best linear combination of “random” vectors

Suppose I have something like: $\vec{v} = \langle 1, 2, 3, 4, 5 \rangle$ and I have a set of vectors (these are all just made up numbers): $\vec{w_1} = \langle 3, 7, -2, -4, 8 \rangle$ $\vec{w_2} ...
1
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1answer
101 views

Show that $V$ is a vector space when $V$ is the set of all continuous functions $\Bbb{S}^1 \to \Bbb{R}$.

Let $V$ (which is infinitely dimensional) be the set of all continuous functions $\Bbb{S}^1 \to \Bbb{R}$. Show that $V$ is a vector space. Define $\langle-,-\rangle: V\times V\to \Bbb{R}$ by $$\langle ...
2
votes
2answers
39 views

Showing set is a vector space

Say I have a set of vectors with multiplication and addition both defined. To prove that it is a vector space I have to confirm the eight axioms. When I check the distributive property for scalar ...
1
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2answers
385 views

Finding a point on a 3d line

I have two points in 3D which will create a single ray. I am trying to find a point on that ray which intersects a plane who's y coordinate is 0. So how do I find a point on a 3D line when I know the ...
0
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2answers
51 views

Find a basis for $x_1,x_2,x_3,x_4$ were $x_2 + 2x_3 + 3x_4 = 0$

In vector space $\mathbb{R}^4$ I'm supposed to find basis for $x_1,x_2,x_4$ from real numbers, while $x_2 + 2 \cdot x_3 + 3 \cdot x_4 = 0 $. I'm not sure how to go about that. Thanks for help, I'd ...
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2answers
43 views

$X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq\varnothing$

Let $X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq \varnothing$ then show that $E$ spans $X$. I am trying it in a following way.... Let be the norm ...
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1answer
279 views

The proof of the infinity base of $\mathbb{R}^{\infty}$

We know that a finite basis of the finite-dimensional space $\mathbb{R}^n$ is $$ \{(1, 0, 0, 0,\ldots,0),\:(0, 1, 0, 0, 0,\ldots,0),\:(0, 0, 1, 0, 0, 0,\ldots, 0),\:\ldots,\:(0, 0, \ldots, 0, 0, 0, ...
2
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1answer
1k views

Orthogonal Projection of a Point onto a Plane

I'm dealing with an exercise that requires I find the orthogonal projection of a given point onto a given plane. I don't want an answer directly for my exercise, I would instead like to understand ...
2
votes
2answers
145 views

Getting angles for rotating $3$D vector to point in direction of another $3$D vector

I've been trying to solve this in Mathematica for $2$ hours, but got the wrong result. I have a vector, in my case $\{0, 0, -1\}$. I want a function that, given a different vector, gives me angles DX ...
2
votes
1answer
52 views

projective geometry and relationship of cross-ratios

Define for pairwise different points $P_i=[v_i]$ the cross-ratio $\operatorname{CR}(P_1,P_2,P_3,P_4) = \frac{\det(v_1,v_2)}{\det(v_2,v_3)}\cdot\frac{\det(v_3,v_4)}{\det(v_4,v_1)}$ on $\mathbb{KP^1}$ ...
0
votes
1answer
42 views

For ${\bf x} = (x_1, x_2) \in R$ find ${\bf y} = (y_1, y_2)$ such that ${\bf x} = y_1{\bf v_1} + y_2{\bf v_2}$

For ${\bf x} = (x_1, x_2) \in R$ find ${\bf y} = (y_1, y_2)$ such that ${\bf x} = y_1{\bf v_1} + y_2{\bf v_2}$ and find a matrix $M \in M_2(R)$ such that ${\bf y} = M{\bf x}$. How do I go about ...
0
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2answers
31 views

How do you rotate a vector by $90^{\circ}$?

Consider a vector $\vec{OA}$. How will I rotate this vector by $90^{\circ}$ and represent in algebraically?
-1
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1answer
43 views

Linear transformation and characteristic polynomial

Let $V $ be an $n$-dimensional vector space and $T : V \to V$ a non-invertible linear transformation. Show that there is a subspace $W \subset V$ which is $(n−1)$-dimensional and contains ...
1
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1answer
49 views

Vector Space Dimension

Let $A,B$ be $n\times m$, $s\times m$ matrices respectively, and let $$V=\{X\in \mathbb{F}^{m\times n};\ B X A=0\}.$$ Suppose that $$rank(A)=r,\ rank(B)=m.$$ Show that $dim V=m(n-r)$. I have no idea. ...
0
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1answer
14 views

A question regarding lines between points.

On pg.13 of Lang's "Second Course in Calculus", the following is asserted: Let $P=(2,1)$ and $A=(-1,5)$. Then the parametric equation of the line through $P$ and in the direction of $A$ gives us ...
1
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0answers
55 views

vector calculus work done line integral

A string lies along the circle $x^2 + y^2 = 4$ from $(2,0)$ to $(0,2)$ in the first quadrant. The density of the string is $ρ(x,y) = xy$. a. Partition the string into a finite number of sub-arcs to ...
1
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1answer
29 views

linear dependence on $\mathbb R^n$

$S_1$ and $S_2$ finite sets on $\mathbb{R}^n$, $S_1$ is a subset of $S_2$, $(S_1\neq S_2)$. If $S_2$ is linearly dependent, so: $S_1$ could be linearly dependent? $S_1$ Could be linearly ...
0
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1answer
81 views

bijective maps leaving cross-ratio invariant are just the projective transformations

Show the bijective correspondence between a. bijective maps $f: \mathbb{P(K^1)} \to \mathbb{P(K^1)}$ which keep the cross-ratio invariant b. projective transformationes, i.e. $f: \mathbb{P(K^1)} \to ...
1
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1answer
115 views

Axiom of choice and vector space bases

Is this: for every vector space $V$, if $B$ and $C$ are bases of $V$, then there is a bijection: $B\to C$ iff the axiom of choice holds true? Or, perhaps, if axiom of choice is replaced by ...
0
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1answer
55 views

Necessary and Sufficient Condition for Vector Space

Problem Assume a finite set $F$, write the necessary and sufficient condition in terms of the number of elements of $F$, such that $F$ is a real vector space. (Assuming that the vector addition and ...
1
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3answers
37 views

Prove that $\mathbb{R}_{\neq0}$ is not a real vector space

Assume a set of $\mathbb{R}_{\neq0}=\{a \in \mathbb{R} \mid a \neq 0\}$, where addition of elements in $\mathbb{R}_{\neq0}$ is the product in scalar $ab$. Prove that this is not a real vector space. ...
2
votes
2answers
165 views

Linear algebra question with dual basis?

I have $B= \{b_1, \dots,b_n \}$ be basis for vector space $V$ over reals. Then if $A = \{a_1, \dots,a_n \}$ be basis for dual space $V^*$ (dual space is defined as set of all linear function mapping ...
5
votes
3answers
338 views

Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?

It is known that any non-diagonalizable matrix, $A$, can be approximated by a set of diagonalizable matrices, e.g. $A \simeq \lim_{k \rightarrow \infty} A_k$. Why this is true? Note: I was faced with ...
1
vote
0answers
116 views

Dual Space Annihilator Question

$T\colon V\to W$ is a linear map, with $V$ being finite dimensional. Show that $\mbox{Im }T'=(\ker T)^0$ where ' refers to dual space and $^0$ refers to the annihilator of a space. My confustion with ...
0
votes
1answer
73 views

If $Ker(T^r)=Ker(T^{r+n})$ does that imply $Im(T^r)=Im(T^{r+n})$

Here T is a linear transformation and an n dimensional vector space V, $T:V\rightarrow V$. And $r$ is such that $Ker(T^r)=Ker(T^{r+j})\ \forall j\geq1$ Also, how can I show that $W=T^r(V)$ is T- ...
4
votes
1answer
82 views

Has anyone succeeded in formalizing the notion of a complete vector space? (Not using topological ideas).

In order theory, we have the concept of a lattice, which is defined as consisting of an underlying set $L$ together with two binary operations $\wedge$ and $\vee$. Now when $L$ is finite, the concept ...
6
votes
1answer
70 views

Linear and Commutative function over Square Matrices.

Find all functions $f$, such that $f(mA+nB) = mf(A) + nf(B)$ and $f(AB) = f(BA)$ , where $A, B$ are square matrices and $ m,n$ are scalars. Need to find $f$ as an explicit function of any general ...
0
votes
1answer
46 views

Finding the change of a basis from 1 matrix to another

I know how to convert a given Basis B to a standard basis. I know how to find a coordinate vector with respect to B given the standard basis. But how do I find the change of basis from a Matrix to ...
2
votes
1answer
69 views

Euclidian Spaces

Let $T$ be a linear operator on $V=\mathbb{R}^n$ whose matrix $A$ is a real symmetric matrix. Could someone show me how to prove that $V=(\text{ker } T) \oplus (\text{Im } T)$ ?
1
vote
2answers
205 views

Orthogonal unit vectors

Let $W$ be a two-dimensional subspace of $\mathbb{R}^3$, and consider the orthogonal projection $\pi$ of $\mathbb{R}^3$ onto W. Let $(a_i,b_i)^t$ be the coordinate vector of $\pi(e_i)$, with respect ...
1
vote
1answer
115 views

Let V be a finite dimensional subspace

1)Let $T : V \to V$ be a nonzero linear operator. Prove that either $T$ is an isomorphism, or there exists nonzero $R$, $S$ in $L(V, V)$ such that $RT = O$ and $TS = O$. 2)Let $I(V)$ be the set of ...
0
votes
1answer
77 views

What is the null vector for the vector space of continuous functions $f \colon \mathbf{R} \to \mathbf{C}$?

The set of all continuous complex-valued functions of real variable $x$ together with addition of two vectors $\boldsymbol{f} = f(x)$ and $\boldsymbol{g} = g(x)$ defined by \begin{equation} ...
3
votes
2answers
173 views

Can you factor out vectors?

My prof introduced eigenvalues to us today: Let $A$ be an $n \times n$ matrix. If there a scalar $\lambda$ and an $n-1$ non-zero column vector $u$, then $$Au = \lambda u$$ then $\lambda$ ...
1
vote
1answer
38 views

Basis for vector space given combination of vector components

The following is the first step in a homework problem of mine: Find a basis for the vector space $S = \{(x,y,z,w) \in \mathbb{R}^4 \mid x - y - 2z + w = 0\}$. The actual problem involves ...
2
votes
3answers
130 views

Two inner products being equal up to a scalar

I would appreciate a hint on the following problem: Let $V$ be a finite dimensional vector space over $F$. There are two scalar products such that: $$ \forall \ w,v \in V \ \Big(\langle ...
1
vote
0answers
56 views

What vector space is this?

Let $a,b,c$ be odd primes. In particular, $ab, ac, bc$ are all odd numbers. We can use this to our advantage, since then $\sqrt[ac]{x} : \Bbb{R} \to \Bbb{R}$ is well-defined and a bijection. Let ...
0
votes
1answer
29 views

If $A\ne V$ is a subspace of $V$ and $B$ is linearly independent subset of $V$, then $B$ can be completed to a basis of $V$

Let $A \ne V$ be a subspace of $V$ and $B$ a linearly independent subset of $V$. Prove that $B$ can be completed to a basis of $V$ with vectors from $V \setminus A$. OK, I started with: ...