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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29 views

Showing Linear dependency

Show that if each of the vectors $\left\{v_1, v_2, . . . , v_n\right\}$ is a linear combination of the vectors $\left\{w_1, w_2, . . . , w_n\right\}$, then $\left\{v_1, v_2, . . . , v_n\right\}$ is ...
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1answer
11 views

Vector Space external direct sum

Question: Give an example to show that it is possible for $A \oplus B = A\oplus B'$ without having $B=B'$, where $A,B,B'$ are subspaces of $_FV$ I really can't imagine this, say let $A \oplus B = ...
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1answer
37 views

Is the empty set a vector in every vector space?

A vector space consists of a nonempty set V of objects. According to set theory, every set must contain the empty set. So I deduce that the empty set is a member of every vector space. But is the ...
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11 views

Spanning Spaces by Different Basis

I have a query related to spanning space by two bases $S_1=\{V_1+V_2, V_3, V_1-V_4,V_3-V_2\}$ $S_2=\{V_1, V_2, V_3, V_4\}$ Can we consider spaces generated by $S_1$ and $S_2$ to be equivalent?? Or ...
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22 views

Relation between the Complex Number System and Vector Spaces

Given that any Complex Number $z$ can be represented as a Vector in $\mathbb{R^2}$ and since a Vector is nothing more than an element of a Vector Space (in its most general form), does that not imply ...
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23 views

$\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert space

I want to describe $\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert spaces; $\mathbb{C}^n\otimes \mathbb{C}^m$ is endowed with the scalar product $\langle x\otimes y, x'\otimes ...
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2answers
26 views

proving n linearly independent vectors are generating

Given n linearly independent vectors $v_1, v_2, ... , v_n$ in a real vector space $V$ where $\mbox{dim}(V)=n$, how do I prove the vectors generate $V$? I understand why this is true, I can visualize ...
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2answers
414 views

Angle between two vectors not in same plane

I want to know how calculate the angle between two vectors and both are not in same plane, which means that they don't intersect at any point? For example how do I calculate angle between AB and EF ...
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1answer
24 views

Forming a basis of P3(R) from a set S.

I seem to have a good understanding of spanning sets and linear independence which then becomes essential for understanding basis, but I am unsure how all this works for the field of polynomials. I ...
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2answers
474 views

How to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$ based on the following assumption?

Suppose $U$ and $W$ are subspaces of a vector space $V$ such that $\dim(U) =\dim(W)$ and $U\ne W$, how to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$? My approach is to use ...
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2answers
22 views

Vector representation

I saw many people representing vectors like this: -----> in a 2D plane. Why do you need the little arrow head over there in the end? Doesn't it make that a ray ...
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0answers
21 views

In a vector space every subspace of codimension $d$ is the zero set of $d$ linear functionals

Let $V$ be a vector space over a field $k$ (not necessarily finite dimensional), and let $W$ be a subspace of codimension $d$, i.e. $\dim(V/W) = d$. Is it true that there exists $d$ linearly ...
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1answer
54 views

Determine a subspace in R^5

I'm having trouble with this question: For (c) obviously I can prove that by the "Closed under scalar multiplication" But the other two I am a bit confused. Hope someone might shed some light on ...
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0answers
22 views

Showing that common p-Norm isn't a norm anymore for $0\lt p\lt 1$

I have the following problem: I need to show that the common p-Norm defined as: $$||.||_p: (v_1, \dots ,v_n) \to (\sum^n_{i=i} |v_i|^p)^{1/p}$$ doesn't constitute a norm on $\Bbb R^n$ for $n \ge 2$ ...
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0answers
85 views

Derivative of Elementwise Function (working on a vector)

I have seen an example (it is in terms of neural network back propagation) that I dont understand. Given: $\textbf{a} = \textbf{x}\textbf{W}_{1}+\textbf{b}_{1} $ (where x is dimension (1x5), $W_1$ ...
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2answers
37 views

Expanding a Proof of Induction on $\Bbb N $ to $\Bbb Q $ (Linear Algebra)

My problem is the following: I have an $\Bbb R$ Vectorspace called $V$ and had to show via induction that $\langle nv, w \rangle=n \langle v, w \rangle$ for $v,w \in V$ and $ n\in \Bbb N$. (it's not ...
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1answer
25 views

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$?

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$? I was thinking about polynomial space and complexe space ...
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2answers
23 views

Find a basis of the subspace of polynomial subspace

Find a basis of the subspace of $${\mathbb R}^3$$ defined by the equation $$-9 x_1 + 3 x_2 + 2 x_3 = 0$$ I'm looking on how to approach this problem since my instructor only showed us how to prove if ...
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1answer
20 views

How to prove that $T(x)=[x]_a$ is surjective?

Let $V$ be a vector space and let $\alpha=\{v_1,\ldots,v_n\}$ be a basis for $V$. Let $T:\ V\to \Bbb{R}^n$ defined by $T(x)=[x]_a$ for every $x\in V$. How to show that $T$ is surjective (onto). Here ...
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1answer
38 views

Geometric intuition for finite vector spaces?

There is a powerful geometric intuition for real vector spaces. Is there any good way of visualizing vector spaces over finite fields?
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1answer
22 views

The Existence of $n-1$ Dimensional Linear Subspace

Let $K$ be an infinite field, $V$ is an $n$ dimensional($n>1$) vector space over $K$. $\alpha_1,\alpha_2,\cdots,\alpha_s \in V$ are non-zero vectors. Proof there exists an $n-1$ dimensional ...
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0answers
8 views

An open ball of a normed vector space is path-connected?

I tried to proove that an open ball of a normed vector space is path-connected. Consider $B(x,\epsilon)$ an open ball of a normed vector space $E$. Take a point $b\in B$. The path connects $x$ and $b$ ...
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2answers
23 views

find a scalar $\alpha$ that satisfy the following

when $v\neq 0$, find a scalar $\alpha$ such that $z:=u-\alpha v$ satisfies $\left \langle z,v \right \rangle = 0$ Is there some trick to this? I tried solving this explicitly and I just ended up ...
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1answer
29 views

Rewrite expression $b^Ty=a^Tx+C$

A really simple question. For four $n$-dimensional vextors $a,b,x,y \in \mathbb{R}^n$ and a scalar $C \in \mathbb{R}$, it is known that $b^Ty=a^Tx+C$. Having another vector $d \in \mathbb{R}^n$, how ...
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0answers
32 views

Scalar multiplication and standard matrix addition

I am working on a problem, and I am asked to determine whether the given set and operations define a vector space, if not indicate which laws fail: $$ V = \left\{ \left. \begin{pmatrix} a ...
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1answer
27 views

Solve the following system of differential equation

Let $A = (a_{ij}\in M_3(R))$ be a real matrix and let $P:= \begin{pmatrix} 0 &1 &0 \\ 0 &1 &1 \\ 1 &1 &0 \end{pmatrix}$ such that $P^{-1}AP = \begin{pmatrix} 1 &0 ...
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0answers
96 views

Let $V$ be a vector space. Prove that the doubleton set $S = \{u, v\}$ is linearly dependent if and only if $u$ is a scalar multiple of $v$.

I prove by contradiction. We assume that $S$ is linearly dependent, but $\vec{u}$ is not a scalar multiple of $\vec{v}$ and show that this leads to something completely and utterly ridiculous! If ...
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1answer
27 views

The dual space of a dual space

Eh, I don't quite understand the first question. Can someone explain it? And for the second question, can I say that they have the same dimension. And since the kernel is ${0}$, it is injective. ...
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1answer
13 views

Vectors in Fields of the form $F^n_p$

I can't understand the concept of vectors in the field $F^n_p$ How many vectors should be there? What should be their elements? For example, if we have a vector space $F^3_2$, {(0, 0, 0), (1, 0, ...
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3answers
20 views

vProve that the $\phi's$ form a basis for $V^*$

Hey guys. I've learned linear algebra before, but I kinda forget the part about dual space. For this problem, I think that because $V^*$ has the same dimension as V, which is n in this problem. And ...
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0answers
26 views

Capacitance and Gauss' Law

If the area of a single plate is $A$, show that the capacitance $C$ = $\frac{q}{v}$ is directly proportional to $A$ but inversely proportional to $d$. You may use Gauss' Law: $\nabla$$\cdot$$E$ = ...
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1answer
48 views

Proving a statement in Linear Algebra

Hi I'm currently studying Linear algebra and I just want to get a few bits straight in my head about it: If I had a vector space - $V$ and $S = \{u_1, u_2\}$ is a subset of $V$ . How can I prove ...
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1answer
56 views

Term for “pre-image, but in a vector-space-y way”?

Consider a linear transformation $T :: V \rightarrow W$. Today I found myself wanting to use "the pre-image of $T$" to talk about "the subspace of $V$ that doesn't get killed by $T$". I guess that ...
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1answer
19 views

Prove that $\dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L)$

Prove that $\dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L)$ for every subspace $\mathbb{F}$ and every linear transformation $L$ of a vector space $V$ of a finite dimension. Theorem: If ...
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0answers
31 views

Infinite extensions of “finite degree under $\mathbb{Q}$” [duplicate]

Consider an algebraic extension $K$ of $\mathbb{Q}$. The degree $[K:\mathbb{Q}]$ of $K$ is defined as the dimension of the extension considered as a vector space. Now, let $\overline{\mathbb{Q}}$ be ...
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0answers
14 views

Prove the existence of unitary operator U and a positive operator P in the following

Let $V$ be a finite dimensional inner-product space, and $T:V \rightarrow V$. Prove that there exists a unitary operator $U$ and a positive operator $P$ on $V$ such that $T = U \circ P$. To be ...
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1answer
1k views

Find the vector equation of a line passing through point A perpendicular to line AB

'Points A and B have coordinates (4,1) and (2,-5) respectively. Find a vector equation for the line which passes through the point A (only the point A), and which is perpendicular to the line AB.' ...
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1answer
30 views

Is $\mathrm{span}\{v_1,v_2,v_3\}=\mathrm{span}\{v_1+v_2,v_1+v_3,v_2+v_3\}$ if $v_1,v_2,v_3\in V$ a vector space over $\mathbb{Z}_2$?

Is $\mathrm{span}\{v_1,v_2,v_3\}=\mathrm{span}\{v_1+v_2,v_1+v_3,v_2+v_3\}$ if $v_1,v_2,v_3 \in V$ a vector space over $\mathbb{Z}_2$? I know that this is true if you assume that $V$ is over any ...
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3answers
53 views

Show that the column space of a matrix is not equal to $\mathbb R^3$

Show that the column space of \begin{pmatrix} 4 &−1& 2 \\ 0 &0& 0 \\ 5 &−1 &6 \\ \end{pmatrix} is not equal to $\mathbb R^3$. I have begun by setting my vector ...
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1answer
43 views

Linear Algebra: showing that $\langle x,y\rangle$ is an inner product on Rn.

Just a simple example: Let $V=\mathbb R^n$, Define $\langle x,y\rangle = x^Ty = x_1y_1+...+x_ny_n$. Verify that the function $\langle x,y\rangle $ satisfies the four conditions of being an inner ...
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1answer
19 views

Basis & Dimension for Joint Subspaces

Assumption: Assume that $V_1$ and $V_2$ are subspaces of $\mathbb{R}^\mathbb{3}$ Question: "Suppose that $V_1$ is the subspace of $\mathbb{R}^\mathbb{3}$ given by $V_1 = \{(2t-s, 3t, t+2s): t, ...
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2answers
50 views

Vector Space, Dimension, Subspaces

Suppose that V is a vector space and dim(V) = 4. W is a subspace of V. Prove directly that W must have finite dimension. My Answer: Since W is a subspace of V, dim(W) must be less or equal to ...
2
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1answer
32 views

Prove that $V = V_1 \oplus V_2$ in the following

Let $V$ be a vector space over $\mathbb{C}$ such that $T^2=1_V$. Define $V_1 = \left\{v\in V |\ T(v) = v \right\}$, $\ V_2 = \left\{v\in V |\ T(v) = -v\right\}$ prove that $V = V_1 \oplus V_2$ For ...
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2answers
127 views

A Pure Maths Approach to Thinking About Vectors

Introduction Generally most students are introduced to the concept of Vector as something that has both a "magnitude and direction" and Scalars as something that only has a "magnitude and no ...
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5answers
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Confused about the definition of subspace

The definition from my textbook is: A subspace of a vector space is a set of vectors that satisfies two requirements: If $v$ and $w$ are vectors in the subspace and $c$ is any scalar, ...
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0answers
20 views

Finding a subspace using combination formula

This is a question from my textbook The answer I have is something like this: let $X_1$ and $X_2$ $\in K$,and a,b $\in R$, because $T(aX_1+bX_2)=aT(X_1)+bT(X_2)=0+0=0$, so K is a subspace of ...
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1answer
49 views

Linear Algebra, Vector Spaces (Subspace Theorem)

Question: "Let F(−∞, ∞) represents the set of all real valued function defined on (−∞, ∞). Using the subspace theorem, show that the set of all differentiable functions on (−∞, ∞) that satisfy ...
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1answer
18 views

Find the dimension of the nullspace and basis of the following $T$

Let $V$ be the vector space of polynomials of degree at most 999 with real coefficients. Define a linear map: $T:V\rightarrow\mathbb{R^{100}}\ $ where $T(p) = (p(1),p(2),......,p(100))\ $ ($p$ ...
2
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1answer
44 views

Subset, Dimension and Vector Space

Suppose that $\{v_1,v_2,v_3\}$ is a linearly independent subset of a vector space $V$, with $\dim(V) = 4$, and that $v_4 \not\in \text{span}\{v_1,v_2,v_3\}$. Prove that $\{v_1,v_2,v_3,v_4\}$ is a ...
1
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1answer
20 views

Vector spaces: understanding a step towards canonical forms

I am studying canonical forms of matrices, and I'm a little stuck on something: Consider a finite dimensional $\mathbb{F}-$vector space $V$ and an endomorphism $\alpha:\;V\to V$, and let $V_{\alpha}$ ...