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 is the rank of the multiplication transformation $A$ if $AX=PX$ and $P$ has rank $m$?

Consider the vector space consisting of all linear transformations on a vector space $V$, and let $A$ be the (left) multiplication transformation that sends each transformation $X$ on $V$ onto $PX$, ...
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44 views

Under what conditions is the operator $Ax = [x, y]x'$ a projection?

Suppose that $V$ is a vector space, $x'$ is a vector in $V$, and $y$ is a linear functional on $V$; write $Ax = [x, y]x'$ for every $x \in V$. Under what conditions on $x'$ and $y$ is $A$ a ...
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Question regarding basis in vector spaces

How can one prove the following proposition ? $ B = (e_{1,...,} e_n )\, $ forms a basis for a space $V$ if and only if each vector of $V$ can only be written as an unique linear combination of ...
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1answer
23 views

Scalar Products equation proof

$\langle \langle x + y, z \rangle \rangle = \langle \langle x, z \rangle \rangle + \langle \langle y, z \rangle \rangle$ It is clear when there are only $\langle \dot \ , \dot \ \rangle$ but what ...
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60 views

Is there any shorthand for $\text{span}\{v_1, \ldots,v_n\}$ which doesn't conflict with any notation in linear algebra?

Some people use $\langle \cdot \rangle $ as a shorthand of $\text{span}$ (e.g. the German wiki), i.e. $$\langle \{ v_1, \ldots,v_n \} \rangle := \text{span}\{v_1, \ldots,v_n\},$$ yet the notation ...
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15 views

Show that the following set is a basis for $M_{2\times 2}$ (the vector space consisting of all $2\times 2$ matrices):

I'm not sure how to test for linearly independence when the vectors are written in this form? As i've only learnt how to when they're written like $v_1 = (1,0,0)$ etc. $$ \left[ \begin{array}{ c c ...
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2answers
39 views

What is the name of the algebraic structure constructed with an abelian monoid and a field?

Take a vector space built from an abelian group $(V,+)$ whose elements are the vectors, a field $K$ whose elements are the scalars, and there is an operation (multiplication by scalars) that ...
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25 views

analysis of complex vector space

null vector in complex space let is vector scalar product of which to itself is zero, for example let us take vector scalar product to itself $(1,i)*(1,i)=1-1=0$ let us consider all null vector ...
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32 views

basis for eigenspace corresponding to eigenvalue that reduces to I?

every single example I ever encountered for finding eignspace basis' always was a situation where the reduced matrix had a null space of 1 or more. but what if after plugging in the eignenvalue and ...
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37 views

$[S]$ means linear span of S. Then $[S] = [[S]]$

My definition of span: Suppose a vector space $(V,+,\cdot)$, and $$S = \{u_1,\cdots,u_n\}$$ (and $S$ is a subset of $V$, not a subspace) $$[S]=:\bigcap_{W\subset V, W\supseteq S} ...
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1answer
41 views

Determine the basis and dimension of vector space

Determine the basis and dimension of vector space generated by $\{u + v + w, v + w + z, w + z + u, z + u + v\}$ where $u, v, z, w$ are linearly independent vectors. What's the best way to do it? I ...
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27 views

Vector Space Basis with Set of Vectors

How might I go about finding the basis of a vector spaced spanned by a set of three vectors? For example, if given the set of vectors {(1,2,3),(4,5,6),(7,8,9)} how ...
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1answer
51 views

A matrix-free way to find a fan basis of $V$?

Let $f:V\to V$ be a linear map, $\dim V =n$. A basis $( v_1, \ldots, v_n)$ of $V$ such that for all $j=1, \ldots,n$ the space $\text{span}(v_1,\ldots,v_j)$ is $f$-invariant is called a fan basis of ...
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3answers
77 views

How to find dimension of vector space

In $\mathbb{R}^5$ there is given vector space $V$. Its dimension is 3. In $\mathbb{R}^{6,5}$ consider the subset $X = \{A \in \mathbb{R}^{6,5} : V \subset \ker A\}$. I have to show that $X$ is a ...
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1answer
14 views

Let $[S]$ be the span of $S$ by the intersection definition, then $S$ subspace $\implies [S] = S$?

My definition of span: Suppose a vector space $(V,+,\cdot)$, and $$S = \{u_1,\cdots,u_n\}$$ (and $S$ is a subset of $V$, not a subspace) $$[S]=:\cap_{w\subset V, w\supseteq S} W$$ ...
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23 views

Subspace given a algebraic condition for the vectors in it

I need to verify if the set: $$U = \{(x,y,z)\in\mathbb R^3|x^2+y+z = 0\}$$ is a subspace or not. However, I don't know how to deal with these algebraic equations that the vectors must satisfy. For ...
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1answer
18 views

linear mapping help

please we need help with something like that: We have given linear mapping $$f: R^3 \to R^2, f([1,2,1])=[1,2], f([1,1,3]) =[0,3], f([2,3,-1]) =[1,1].$$ We need to know: $$ f([0, 2, 5]) = [?,?]$$ ...
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2answers
28 views

Subspace of Vectors

Let $W_1$ and $W_2$ be subspaces of a vector space $V$, Show that $W_1\cap W_2$ is a subspace of $V$. Let $W_1+W_2=\{w_1+w_2 \mid w_1 ∈ W1, w_2 ∈ W_2\}$. Show that $W_1+W_2$ is also a subspace of ...
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1answer
18 views

Prove two sets span the same subspace

I've found here that in order for two sets to span the same subspace, the following must be true: Each vector in S1 can be written as a linear combination of the vectors in S2; and Each vector in S2 ...
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0answers
17 views

How to use three complex vector components to calculate resultant complex vector

This is a practical problem related to complex vectors. Imagine you want to find the resultant electric field of multiple electromagnetic waves that have parallel and perpendicular components ...
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1answer
34 views

Proof that the span of the empty set is the $0$ vector

Well, my definition of span is the following: Span: Suppose a vector space $(V,+,\cdot)$, and $$S = \{u_1,\cdots,u_n\}$$ (and $S$ is a subset of $V$, not a subspace) ...
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2answers
65 views

Show that $\lim_{p→∞} ||x||_p = ||x||_∞$

For any $x ∈ \mathbb{R}^n$ and $p ≥ 1$, define $$||x||_p = \left(\sum_{i=1}^n|x_i|^p\right)^{\frac1p},||x||_∞ = \underset{1≤i≤n}{max}|x_i| $$ Show that $$ \underset{p→∞}{lim} ||x||_p = ||x||_∞$$ ...
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2answers
42 views

Why square matrix with zero determinant have non trivial solution

While I was solving problem to determine if given set of vectors are linearly independent or not, the solution given was that the coefficient matrix A of the homogeneous linear system of those vectors ...
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1answer
43 views

Prove that the set, C, of continuous functions is not a subspace of the differentiable functions.

I cannot for the life of me figure out this problem: Prove that the set, C, of continuous functions on the interval (-1, 1) is not a vector subspace of the set, D, of differentiable functions on that ...
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1answer
50 views

Checking whether a vector is in the span of a set of vectors

Suppose we have a set of vectors $S = \{u,v,w\}$ in $\mathbb{R}^3$. We want to find if some vector $x$ is in the span of $S$. From what I understand, for $x$ to be in the span of $S$, we need to come ...
2
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2answers
42 views

Alternative definition for span and proving it is equivalent to the most common one

This is a question related to something that I asked here about this alternative definition of span. User hardmath has helped me a lot! Therefore, I can't still understand how to prove the equivalence ...
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1answer
17 views

Creating a Basis

My question is to find a basis for $S$ when $S = \{ \langle a,b,c,d \rangle \mid a,b,c,d \in \mathbb{R} , a=c, d=a+b \} $. I'm struggling creating a set of vectors that is linearly independent with ...
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2answers
34 views

Proof that the sum of two subspaces $A$ and $B$ when $A\cap B = \{0\}$

If $$A\cap B = \{0\}$$ and $A$ and $B$ are subspaces, then the sum of the subspaces is defined by $$A+B = \{a+b |a\in A, b\in B\}$$ If their intersection is $\{0\}$ then its natural to think that ...
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1answer
51 views

Strange definition for finitely generated subspaces

My teacher gave a definition of a finitely generated subspace $[S]$. I don't even know what does this mean and why it's useful to define, but he said that: Suppose a vector space $(V,+,\cdot)$, and ...
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2answers
17 views

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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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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3answers
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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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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3answers
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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
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
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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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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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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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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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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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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24 views
+50

Co-planar unit vectors and their resultants.

Let $\bf a,b,c$ be three co-planar unit vectors such that $\bf a\hat \;b=\theta,b\hat\;c=2\theta$ where $\bf x\hat\;b$ is the angle between two vectors. Two vectors $\bf p,q$ are given by $$\bf ...