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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Determine whether S is a subspace of P3. Vector space of all real polynomials.

ATTEMPT: Have given a small attempt just really confused on how to approach. So I got the general equation of $p(x)= a + bx +cx^2 +dx^3$. So we find the derivative? and find the values of ...
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0answers
128 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
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2answers
3k views

matrices forms a basis for vector space 2x2

$\begin{bmatrix}0&1\\2&3\end{bmatrix}$ $\begin{bmatrix}3&4\\5&6\end{bmatrix}$ $\begin{bmatrix}7&8\\9&10\end{bmatrix}$ $\begin{bmatrix}11&12\\13&14\end{bmatrix}$ Show ...
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2answers
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Is there a name for the set of bit combinations of bitstrings?

Let $A \subset \{0,1\}^n$ be a set of $n$-bit bit vectors. Let me call a bit vector $b = (b^{(1)}, b^{(2)}, \dotsc, b^{(n)}) \in \{0,1\}^n$ a "bit combination" of the vectors in $A$ if: $$\forall i ...
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2answers
66 views

A set of n generators of a subspace of dimension n.

A set of $n$ linearly independent vectors in $n$-dimensional subset $V$ IS a basis of this subspace. But what about a set of $n$ generators in this subspace? Is it a basis of $V$ for sure?
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1answer
124 views

Show that a normed Vector space is complete, need smart help.

I want to show that a normed vector space is complete. I know that if you can show that every Cauchy sequence converges, then it is complete. But in a normed vector space, completeness is equivavlent ...
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1answer
95 views

A generating subset of a vector space contains a basis

Let $V$ be a vector space having dimension $n$, and let $S$ be a subset of $V$ that generates $V$. Prove that there is a subset of $S$ that is a basis for $V$. (Be careful not to assume that $S$ is ...
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1answer
116 views

Where is piecewise dirichlet function with $|x|^2$ continuous or differentiable?

If $|x|^2$ is continuous and differentiable on all of $\mathbb{R}^n$ (already shown differentiability by showing all $n$ of its partial derivatives are continuous), then... Question: For the function ...
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2answers
326 views

Finite dimensional subspace of $C([0,1])$

Let linear $S$ be a subspace of $C([0,1])$, i.e., the continuous real-valued functions on $[0,1]$. Assume that there exists $c>0$, such that $\|\,f\|_\infty\leq c \|\,f\|_2$, for all $f\in ...
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1answer
39 views

How to show a subset doesn't span a space?

Given that $\{v_1,…,v_m\}$ is linearly independent, how do you show that $\{v_2,…,v_m\}$ does not span that same vector space?
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2answers
47 views

Is $S$ a subspace of $V$?

Let $V$ be the set of real-valued continuous functions on the interval $[-3, 3]$. $S$ is set of real-valued functions with condition $f(-1) = f(1)$. Is $S$ a subspace of $V$? Prove, and if not, why?
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0answers
125 views

Norm inequality with wedge product

Anyone could help me to prove this following inequality? $\displaystyle\frac{||(u+v)\wedge w||}{||u+v||}\le \frac{||u\wedge w||}{||u||} +\frac{||v\wedge w||}{||v||} $ where $u\wedge v$ is the wedge ...
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1answer
70 views

Homework excercise, completeness in Vector-spaces, is it correct?, long, but can it be simplified?

I have a very difficult excercise. I see now that it became too much text for someone to might go through it, if you can please help me, but don't want to read all, can you please then only answer my ...
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2answers
99 views

Hilbert vs Inner Product Space

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product. Also can a metric always be defined ...
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5answers
1k views

Why is one proof for Cauchy-Schwarz inequality easy, but directly it is hard?

Let's say you are in $\mathbb{R}^n$ and you define the norm as $||x||=\sqrt{x_1^2+x_2^2...+x_n^2}$. This we recognize as the usual norm from the inner product: $||x|| = \sqrt{\langle x, x \rangle}$, ...
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2answers
245 views

Intersection of two subspaces in 4D

I would like to know if there is some way to imagine the case when a 3D subspace intersects with a 2D plane in a 4D space. For example, let's have a 3D space in 4D $$A = \left(\begin{array}{c}1 & ...
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1answer
75 views

What is the relation between basis vectors of a vector space to those of its subspace?

From this question: Suppose $V$ is a vector space with dimension $6$. Let A and B be subspaces of V with dimensions 4 and 5 respectively. What are the possible values for the dimension of A ...
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0answers
128 views

Books which explain vector analysis/algebra in detail.

I'm trying to learn vectors but I can't find a decent book which explains vectors in depth. I need a book which explains vectors from the beginning, using a beginner's approach(assuming the reader ...
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1answer
906 views

How to use geometry to express unit vectors of spherical coordinate system in terms of Cartesian unit vectors

It's quite easy to express unit vector $\hat{r}$ in sum linear combinations of Cartesian unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$. But I am not sure how I can use geomtery to find a Cartesian ...
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1answer
47 views

Vector space basis

If I have no fundamental misunderstanding of vector spaces, my question is as follows. If an orthogonal basis of a vector space consists of $N$ vectors, is this right that every vector from this ...
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2answers
80 views

Cartesian & Tensor Product

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?
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2answers
179 views

Question about Normed vector space.

Here is the definition of a normed vector space my book uses: And here is a remark I do not understand: I do not understand that a sequence can converge to a vector in one norm, and not the ...
3
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0answers
45 views

Find closest vector to a given vector from a particular set of vector

Let $x=\left(x_t\right)_{t=1}^n$ be a vector such that $$ x_t = \prod_{i=1}^t u_i, \tag{1} $$ where each parameters $u_i$ can take any of two value $$ u_i \in \left\{a,b \right\} = \left\{ 1.3, 0.8 ...
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1answer
133 views

If $U$ is a subspace of $V$, there exists $W$ such that $T:V\to W$ has $ker(T)=U$.

I am having trouble working out a proof for this question, is it something to do with $U$ and $W$ being complementary subspaces? I cannot find a way to prove that there will always exist a $W$ for all ...
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0answers
78 views

Zero dimensional ideals and their primary decomposition

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring over a field $k$, and $I$ a zero dimensional ideal with a primary decomposition $I=\cap Q_i$. Why is $\sum \dim_k S/Q_i = \dim_k S/I$?
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2answers
75 views

Existence of a linear transformation in an infinite dimension vector space.

If $V$ and $W$ are vector spaces, $\beta=\{v_1, \ldots , v_n\}$ is a finite a basis for $V$ and $\{w_1, \ldots , w_n\}\subset W$, we know there is an unique linear transformation $T:V\rightarrow W$ ...
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2answers
3k views

Do four dimensional vectors have a cross product property? [duplicate]

We know how to make cross product of three dimensional vectors. $$ \vec A \times \vec B = \vec C$$ Where : $ \vec A = (A_i; A_j; A_k)$ $ \vec B = (B_i; B_j; B_k)$ $ \vec C = (C_i; C_j; C_k)$ $C_i = ...
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2answers
587 views

Is the set of all singular matrices under standard operations a vector space?

This question is very difficult for me to visualize. It asks me to determine whether the below is a vector space and if not, what axiom it fails: The set of all $2\times2$ singular matrices with ...
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2answers
66 views

Is this a vector space?

I'm asked to identify if the following is a vector space and if it is not I need to identify the axiom it fails. $\begin{bmatrix} a & b \\ c & 1 \end{bmatrix}$ I don't think it is a vector ...
3
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1answer
55 views

Finding the dimension of a vector subspace

Consider $\mathbb{F}_{2}^{n} = \{(k_{1}, k_{2}, ... , k_{n}) : k_{i} \in \{0,1\}$ mod $2\}$. Let $M$ be the subset of $\mathbb{F}_{2}^{n}$ given by $k_{1} + k_{2} + \cdots + k_{n} = 0$. Prove that ...
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1answer
30 views

how to do these vectors? need more details

Let $v_1$, $v_2$, $v_3$ be mutually orthogonal non-zero vectors in 3-space. So, any vector $v$ can be expressed as $v=c_1v_1+c_2v_2+c_3v_3$. (a) Show that the scalars $c_1$, $c_2$, $c_3$ are given by ...
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1answer
25 views

Get confused with scalar projections

Use scalar projections to find the distance from the point (−2,3) to the line 3x−4y +5 = 0. so |-2*3+3*-4+5| / sqrt(3^2 + -4^2) just want to know if the equation is correct? anyone plz verify it?
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0answers
20 views

Checking convexity by looking at 2-dimensional cross-sections

If I have a closed set of n-dimensional points and I want to know if it's convex just by examining some set of 2-dimensional cross-sections (and checking each cross-section for convexity), how small ...
1
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0answers
45 views

permutation polynomial

If we have GF(4) as an extension field, we can define a permutation polynomial of GF(4) like L(x), a linearized polynomial, of the followinf form: L(x)= \sum_{s=0}^{\r-1} a_s x^(q^r)e Is it possible ...
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2answers
43 views

How to see a function as a vector in a vector space

I know that strictly speaking my question is some sort of a duplicate of at least this previous one and I am quite sorry for that (usually I try to get the best from previous questions), but still I ...
0
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1answer
37 views

vector problem, get very confused

Let $v_1$, $v_2$, $v_3$ be mutually orthogonal non-zero vectors in $3$-space. So, any vector $v$ can be expressed as $v= c_1v_1 + c_2v_2 + c_3v_3$. Show that the scalars $c_1$, $c_2$, $c_3$ are ...
2
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1answer
85 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
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1answer
241 views

Quadruple product

Looking to prove the following: $\langle a\times b,c\times d\rangle =\langle a,c\rangle \langle b,d\rangle -\langle a,d\rangle \langle b,c\rangle$ Where $\langle ,\rangle$ and $\times$ denote the ...
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1answer
28 views

multivariable calculus question norm

Given vector space C([a,b],$ \mathbb{R} $) of continuous functions of [a,b] in $ \mathbb{R}. $ Prove that the function $ \left \| f \right \|_{1}=\int_{a}^{b}\left | f(t) \right |dt $ is a norm. Also ...
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1answer
35 views

Problem with plane and angles

I have the non coplanar straight lines that touch in (1, -2, 3): $$L1: \frac{x - 1}{2} = \frac{y + 2}{2} = \frac{z - 3}{1}$$ $$L2: \frac{x - 1}{3} = \frac{3 - z}{-4}; y = -2$$ $$L3: \frac{x - 1}{2} ...
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1answer
42 views

Straight lines forming an equilateral triangle

I have the straight lines: $$L1: (1, 0, 0) + r(1, 1, 1)$$ $$L2: (7, 4, 3) + s(3, 4, 2)$$ I'm asked to get the vertices of the equilateral triangle of side 2 * 2 ^ (1/2) so one vertex belongs to L2 ...
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50 views

Basis for the power vector space of a vector space

Let $V$ be a vector space over a field $F$ , let $P(V)$ denote the power set of $V$ ; for $A, B \in P(V) $ and $ a \in F$ , define $A +' B :=${$x+y : x\in A , y\in B$ } and $a.A:=${$ax : x\in A$ } , ...
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3answers
52 views

Determining if a point is inside two planes

I have two planes(Plane 1 and Plane 2) the normals ($n_1$ and $n_2$) of which are known to me. How do I determine if a point is inside the two planes? By inside I mean the 3d space between Planes 1 ...
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0answers
33 views

Have another question about straight lines

I made a question about this topic some hours ago and i found another problem that i can't solve. I hope this is not against the rules of "homework". So i have the following lines: $$L1: P1(1, 1, 2) ...
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1answer
40 views

I have some problems with straight lines and planes

Firstly, I need to say that English is not my first language and the problems were written in Spanish. I have never read a Math problem in English, so some words may be confusing. If they are, please ...
3
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2answers
291 views

Finding a basis for a given subspace of $\Bbb R^4$

Find a basis for the subspace $ W = \{(x, y, z, w) \in\Bbb R^4 : y − 2z + w = 0\}$. What is $\dim(W)$? I don't seem to understand how to solve this problem. I just don't know where to start I am not ...
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1answer
361 views

A trivial solution vs. a non-trivial solution - involving vectors

I'm not entirely sure I understood this question in my text book, but it said the following: The zero vector $0 = \left(0,0,0\right)$ can be written as a linear combination of the vectors $v_1$, ...
2
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2answers
76 views

dimension of that vector space?

I would like to know how to prove that the dimension of $L(E,E)$ (that is the set of linear maps from $E\rightarrow E$) where $E$ is a finite dimensional vector space ($dim E=n$). I know that the ...
0
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2answers
71 views

Multivariable calculus - find total derivative

I want to find the total derivative of the function $f: \mathbb R^n \to \mathbb R^n$, $f(x)=\frac{x}{|x|}$ If I was to copy what the teacher taught, I should find the limit of $\lim_{t \to 0} ...
1
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2answers
57 views

How to prove that equality?

Let $E$ be a vector space of finite dimension and $f:E\mapsto E$ be a linear map (that is $f$ is an endomorphism) such that $(f\circ f \circ f) (E)=f(E)$. I want to prove that $E=f(E)\bigoplus ...