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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Vectorspace set as dimension

I encountered some notation in my mathematics exercises which I couldn't make sense of and couldn't find on the internet. Usually, a vector space is written like this: $K^n$. For example, ...
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2answers
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Coplanarity of two lines in 3D

Suppose we have 2 lines $$l_1 : x = 5 , \frac{y}{3-\alpha}=\frac{z}{-2}$$ and $$ l_2: x= \alpha , \frac{y}{-1}= \frac{z}{2-\alpha}$$ so what will be value of $\alpha$ for lines to be coplaner ? I ...
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42 views

Nontrivial polynomial relation for real $n \times n$ matrices?

I am going through Michael Artin's Algebra book to brush up on concepts and question M.2 at the end of chapter 3, Vector Spaces, struck me as odd: Let $A$ be a real $n\times n$ matrix. Prove that ...
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3answers
2k views

Show that the area vectors for a general tetrahedron sum to zero

Using vector addition and multiplication, it is possible to show that the sum of the area vectors for a general closed tetrahedron in $\mathbb{R}^3$ (3-space) is zero. Hint: start by writing down ...
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1answer
27 views

Functions in $\mathbb {R}[X] $

For the ring of polynomials over the reals, which can be considered an infinite-dimensional vector space with infinite monomial basis, is the following true: Any analytic function $f$, which is ...
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2answers
28 views

Linear independence of a set of mappings

$Map(\mathbb{R},\mathbb{R}):=$ The set of all mappings from $\mathbb{R} \rightarrow \mathbb{R}$ For every $a \in \mathbb{R}$ there is a Funktion $f_{a}:\mathbb{R} \rightarrow \mathbb{R}$ with: $$ ...
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1answer
281 views

Find normal vector of circle in 3D space given circle size and a single perspective

I don't really know what to search up to answer my question. I tried such things as "ellipse matching" and "3d circle orientation" (and others) but I can't really find much. But anyways... I have ...
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1answer
30 views

Geometric interpretation of polynomial space

For example $p_3$ has ${1,t,t^2,t^3}$ bases. Is their any geometric interpretation for it? In addition,I have seen that the polynomial function $g(t)$ is considered as a vector! And the sum of two ...
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1answer
31 views

What is the name of this transformation?

I have: * Set of $10$ integers: $\operatorname{Set}={1,2,3,4,5,6,7,8,9,10}$ * $\operatorname{2D}$ vectors with integers from the set; for example: $x=(2, 6)$ A integer can't appear twice within a ...
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0answers
26 views

Trying to prove any symmetrical matrices would be a vector space

Here is the question I am struggling with; Let $V$ be the set of any real symmetric matrices, that is, the set of all matrices $A$ such that $A^{T}=A$ For whatever reason I just can't seem to find ...
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1answer
17 views

Prove a dieudonne vector space is a metric space

I'm supposed to prove that a dieudonne vector space is a metric space, but I'm stuck on the triangle inequality. I need to show that $d(x,z) \le d(x,y)+d(y,z)$ with $d(x,y)=|x-y|= \sqrt{(y-x|y-x)}$ ...
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1answer
27 views

Properties of Determinants in True or False Questions

These are some good practice problems for anyone searching on the Web for determinants problems. There is one or two questions that I am not getting right according to the system. Could you help me ...
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1answer
28 views

Difference between sum of vector spaces and union of subspaces?

I'm having trouble understanding the difference between summing two subspaces and making ther union. My book says that the sum of two subspace is also a subspace, but I've found this example that ...
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2answers
22 views

True/false questions on image kernel and basis of vector spaces and subspaces.

1) The set ${t + 1, t2 + 2, t2 + t}$ is a basis of $F_3[t]≤2$. I put false because if t is 2, then we have ${t + 1, 0 , t2 + t}$ so a non zero coefficient could exist. 2) T : V → V a linear ...
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2answers
24 views

Find a basis of the image of a linear transformation defined by: $T(a, b, c, d) = a(1 + t + t ^2 ) + b(t + t^ 2 ) + ct^2 + d$.

$T(a, b, c, d) = a(1 + t + t^2) + b(t + t^2) + ct^2 + d$ is a linear transformation. I have no idea how to go about this. Is there a way to do it without using matrices? $T: C_4 → C[t]_{≤2}$ ...
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0answers
27 views

A homomorphism from a finite dimensional vector space into itself has a non-trivial nullspace

I've been asked to prove the following claim: If V is finite dimensional and $f$ is a homomorphism of V into itself, there is some $v\neq0$ such that $f(v)=0$. However, this does not seem true ...
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1answer
22 views

C-vector space V linear transformation T: V → V . Show that the image + kernel is a direct sum.

A linear transformation of C-vector space (complex field) where $T: V → V$ and $T ◦ T = −2T$. $$\dim(V) = n$$ How can we prove that $\operatorname{Im}(T) + \ker(T)$ is direct? I know that i have ...
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2answers
28 views

True false about direct sum and their bases of vector spaces

I am not entirely sure about the following true/false questions For all the following : $V$ a vector space and $W_1$ and $W_2$ two subspaces such that $V = W_1 ⊕ W_2$ 1) for all subspaces U of V : ...
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2answers
22 views

Is the $0$ vector of a linear subspace the same as the $0$ of the vector space?

I'm asking this because I'm trying to prove that $P_s(\mathbb R)$ is a linear subspace of $P_n(\mathbb R)$, where $$P_n(\mathbb R) = a_0x^0 + a_1x^1+\cdots + a_nx^n$$ If $P_s(\mathbb R)$ is a linear ...
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0answers
11 views

Show that the subspace $B_1$ is a basis of $C^4$

I have $B_1$ = $((i,0,0,0),(1,0,1,i),(0,2,i,0),(-i,0,0,i))$ And $C^4$ is a vector space and a basis of it is $C_b$ = $(e_1,e_2,e_3,e_4)$ I want to show that $B_1$ is a basis of $C_b$. So i introduce ...
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0answers
15 views

Dimension of $C(M)$

Suppose that $$M=\begin{pmatrix} 1 & 0 & 0 & . & . & 0 \\ 0 & 2 & 0 & 0 & . & . \\ 0 & 0 & . & 0 & . & . \\ . & 0 & . & . & ...
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2answers
24 views

Vector space, set that generates vector space

I have difficulties with this problem: V is a set of all real matrices 2 x 3 such as that the sum of elements in the matrices is equal to zero. The set V with addition and scalar multiplication ...
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1answer
42 views

Is this a basis of $\mathbb{R^4}:$ $\;(1,1,-1,2),\; (1,0,1,-1),\; (2,-1,1,-1)\;\;?$

Do the vectors below form a basis of $\mathbb R^4\;?$ $$(1,1,-1,2),\; (1,0,1,-1),\; (2,-1,1,-1)$$ I tried to represent them as a linear combination equaling zero, and the equations arose from it. ...
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1answer
24 views

Basis of column/row space of $A$: using pivot columns of $A$ vs. $\text{rref}(A)$?

When we have column vectors and want to check which ones are linearly dependent to take them out and form a basis for the column space of $A$, we put them as column vectors in the matrix. Then, we ...
0
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1answer
23 views

Condition for dim of the Euclidean space with orthogonal basis

I would like to show that if the orthogonal basis of the $\Bbb R^n$ Euclidean space with the standard dot product has the vectors whose elements are exclusively $1$ or $-1$, then $n \le 2$ or $n$ is ...
0
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1answer
25 views

how am I supposed to do these problems any differently? (finding basis for row space)

I'm given two problems, which look exactly the same except the second one says "consisting of only row vectors of A". here are the problems: on 5 II, I ended up row reducing and writing my basis ...
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1answer
236 views

Finding coordinate vector

The coordinate vector of: $$ \begin{pmatrix} 1&2&-1\\0 & 0 & 6 \end{pmatrix} $$ with respect to the standard basis of $M2x3$ would be: Are you just finding the inverse of this ...
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1answer
29 views

Kernel and Range of a linear transformation

So the question is let T:M2x2 -> R be defined by T(A) = tr(A). Find bases for the kernel and range of the linear transformation T. Could someone explain how to solve this as I don't quite understand ...
2
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3answers
71 views

Basis for a vector space consisting of the set of linear combinations of these functions

S is the vector space consisting of the set of all linear combinations of the functions $f_1(x)=e^x$, $f_2(x)=e^{-x}$, $f_3=sinh(x)$. Find a basis for S and find the dim[S]. First, I let ...
0
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3answers
31 views

How to show vectors are linearly independent

How do I show these vectors are linearly independent in $\mathbb{R^2}$ . If they are not, what would be the dependency relationship? $(2,-1), (3,2), (0,1)$? Wouldn't they be linearly dependent? If ...
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1answer
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Is this a basis for the set S?

Consider the set: $\begin{bmatrix} a & b \\ -b & a \\ \end{bmatrix}$ Would the basis be found by doing: a$\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$ + $b\begin{bmatrix} 0 ...
1
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3answers
80 views

Prove that $\mathrm{span}\{ I,A,A^2… \} = \mathrm{span} \{ I,A,A^2,…, A^{k-1}\}$

Let $A\in M_n(F)$ and $k=\deg(m_A)$ where $m_A$ is the minimal polynomial of $A$. Prove that $\mathrm{span}\{ I,A,A^2... \} = \mathrm{span} \{ I,A,A^2,..., A^{k-1}\}$ So we have that $m_A = a_0 ...
0
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4answers
54 views

Show that U ∩ V $\neq$ {0}.

This is the question I am trying to solve: Suppose that U and V are 4-dimensional subspaces of $\mathbb{R}^6$ and that dim(U+V)=6. Show that U ∩ V $\neq$ {0}. I don't know how to approach this ...
5
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2answers
1k views

Is the zero vector in $\mathbb{R}^n$ by itself a subspace of $\mathbb{R}^n$?

W is a subspace of $\mathbb{R}^n$ iff The zero vector ∈ W. X + Y ∈ W for any X, Y ∈ W. aX ∈ W for any X ∈ W and a ∈ R. So, given W = { X : X = [x1...], x1 = 0, x2 = 0, ... xn = 0 } ∈ Rn The zero ...
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3answers
33 views

Do these vectors span $\mathbb{R^2}$?

How do I show that $V_1 = (1,1)$ and $V_2 = (-1, 2)$ , $V_3 =(1,4)$ span $\mathbb{R^2}$? I thought you can express them as a linear combination equaling some fixed $x_1$ and $x_2$, but then I get ...
1
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1answer
32 views

How to find a basis for the solution space of a linear system?

How to find a basis for the solution space of this linear system? $$ \begin{bmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$ Solutions ...
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3answers
34 views

Linear combinations and spanning solution spaces [closed]

PLEASE help on part A only :) V = Column vectors [1 0 -1] and [1 3 0]. a) Make a system of 3 unknowns and 3 equations of which he solution space is spanned by V? b) Express [1 2 3] as a linear ...
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1answer
31 views

Find the flaw - Sum of two subspaces is a subset of their union

Here is a flawed proof for $V+W\subseteq V\cup W,$ where $V$ and $W$ are subspaces of $\mathbb{R}^n$: Consider $\mathbf{x}\in\left(V\cup W\right)^\perp.$ This implies: ...
13
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6answers
344 views

Other guises for the vector space $\mathbb{R}^n$?

One way the vector space $\mathbb{R}^n$ can come up is as the space of polynomials over $\mathbb{R}$ of degree at most $(n-1)$ . Here we have the isomorphism: $$(a_0,a_1,\ldots,a_{n-1}) ...
0
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1answer
31 views

$r(S+T)\le r(S) + r(T)$?

If $V$ is a finite-dimensional vector space, and $S$ and $T$ are linear transformations from $V$ to $V$, how can you show that $\text{im}(S+T)$ is a subset of $\text{im}(S) + \text{im}(T)$ and also ...
0
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2answers
18 views

If a composition of linear transformation is invertible, then are each linear transformations invertible?

If $S$ and $T$ are linear transformations from set $V$ to $V$, which is a finite-dimensional vector space, and if the composition $ST$ is invertible, how can we show that $T$ is one-to-one, therefore, ...
1
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1answer
17 views

Find a basis where vector has fixed cooridnates

What's the method for finding basis where $\beta_1=(0,2,1) \ , \ \beta_2=(1,1,2)$ and $\beta_1$ has cooridnates $1,2,-1$ and $\beta_2$ has $0,0,1$ ? we have that $(1,1,2)=(a_3,b_3,c_3)$ and ...
0
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1answer
32 views

prove linear independence of polynomials

Let $f_1,...,f_k$ be polynomials at field $K$ not including the zero polynomial and $\deg f_i \neq \deg f_j$ for every $i \neq j$ Show that polynomials $f_1,...,f_k $ are linear independent. I don't ...
3
votes
1answer
284 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
1
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1answer
26 views

Proving or disproving the existence of a vector space

Suppose the vector space $V$ is spanned by $(1,0,1,0), (0,1,0,1), (1,1,0,0)$. Is it possible to find a subspace U of $\mathbb R^4$ such that $V \subsetneq U \subsetneq \mathbb R^4$? Note that ...
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2answers
62 views

Find the projection p of x onto the span of u1 and u2

where $u_1=(2/3, 2/3, 1/3)$ and $u_2=(1/\sqrt2, -1/\sqrt2, 0)$ and $x=(1,2,2)$ how do I find the span of $u_1$ and $u_2$? after that do I just use the formula for the vector projection of x onto the ...
0
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0answers
21 views

If the Unit Vectors are equal, Are the directions equal too?

Given that the unit vector of x = unit vector of y, can we conclude that the Direction (or Sign) of x is always equal to Direction of y?
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votes
3answers
34 views

Finding subspace's base

Let W be a subspace of $\mathbb{R}^4$: $ \begin{cases} x_1+2x_2+3x_3+4x_4=0 \\ 2x_1+2x_2+x_3+3x_4=0 \end{cases}$ Find base of W and extend it to the base of $\mathbb{R}^4$ How to approach this ...
0
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0answers
9 views

calculating position of a point knowing two reference lengths

Hi, I would like to know if there is a way to calculate a unique position for Point A knowing the lengths l1 and l2 which are variable string lengths. Point A can move within the range shown below. ...