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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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
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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
38 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
39 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
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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 ...
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0answers
33 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
48 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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2answers
45 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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3answers
229 views

Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces?

Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces? I am having a very hard time at digesting tensor products ...
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1answer
27 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
207 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} ...
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1answer
70 views

paper about linear independence in altered Vandermonde and Cauchy Matrices

Both Vandermonde and Cauchy matrices with $n$ rows and $k$ ($n \geq k$) columns have the property that any $k$ rows are linearly independent (assuming the coefficient are independent). It seems to me ...
3
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2answers
147 views

if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$? [duplicate]

If $A$ is abelian group and $B$ is a subgroup of $A$, $B$ is normal subgroup of $A$. Is it true that $B \times A/B \cong A$? I ask because I was watching an online lecture from a course in abstract ...
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1answer
47 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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2answers
43 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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3answers
41 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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2answers
35 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
38 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 ...
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1answer
20 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
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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 ...
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0answers
27 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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1answer
63 views

Linear Algebra Questions over vector Spaces

I have 3 questions and I am wondering whether or not the following are always true or never true. 1.If the null space of 5x4matrix is 2D, then the column space of the matrix can be isomorphic to a ...
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2answers
33 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 ...
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1answer
36 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 ...
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1answer
49 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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0answers
11 views

Find an equation of the plane through the line of intersection…

Find an equation of the plane through the line of intersection of the planes x - z = 1 and y + 2z = 3 and perpendicular to the plane x + y - 2z = 1. Is this set up right? x y z 1 0 ...
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1answer
36 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
199 views

Subspaces, transformation matrices exercise

I have trouble understanding the following exercise so I would really appreciate any help you could give me: Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Let $H$ be ...
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3answers
77 views

Basis on vector space $V$

If $S_i$ is a set of linearly independent vectors of vector space $V$ and $S_g$ a set of generators of $V$. Prove that it exist $S'_g\subset S_g$ that $S_i\cup S'_g$ is a basis of $V$. Notice that ...
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1answer
14 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
34 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
26 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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0answers
32 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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0answers
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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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1answer
29 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 ...
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2answers
53 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
61 views

Why each nonempty weakly open set of an infinite dimensional normed linear space is unbounded with respect to the norm

Suppose $V$ is an infinite dimensional vector space, $f_i$ ($i$ is from $1$ to $n$) are real-valued linear functions on $V$, I cannot understand why the intersection of kernels of $f_i$ must contain ...
2
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2answers
65 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 ...
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1answer
30 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 $v_1$, ...
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2answers
46 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} ...
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2answers
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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 ...
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1answer
28 views

Direct sum of $3$ subspaces

$V_1$,$V_2$,$V_3$ are subspaces of vector space $V$. How to prove that if $V_1 \cap \left(V_2+V_3\right) = V_2 \cap \left(V_1+V_3\right) = V_3 \cap \left(V_2+V_3\right)=\{0\}$ so $V_1\oplus V_2 ...
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1answer
54 views

What is transition matrix

Every e_j har coordinates in the first base: $$e_j = \sum_{i} s_{ij}e_i $$ so writing those coordinates as column vectors we get an important transition matrix $S = (s_{ij})$ and Theorem: ...
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0answers
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Find $y$ where 3 vectors appear on one line

For my homework I need to find the following: There is a house with 3 people looking out of their window. $p1 = (11,10,4), p2 = (12,7,2), p3 = (13,12,3)$ There is an elevator that moves along the ...
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3answers
504 views

Proving Vector Subspaces

Question 1: The set $\mathbb R^3$ of all column vectors of length three, with real entries, is a vector space. Is the subset $$B=\{xyz \in \mathbb R^3|xy+yz= 0\}$$ a subspace of $\mathbb R^3$? ...
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2answers
19 views

If the scalar triple product of three vectors is zero, must the plane they lie in pass through the origin?

$[x,y,z]=0$ implies that the three vectors are coplanar but I don't see why it has to pass through the origin. Are there any counterexamples for this?
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0answers
38 views

Let $V$ be a vector space over field $C$ with inner product $<.,.>$ prove

Prove that $$4(x,y) = |x+y|^2-|x-y|^2+i|x+iy|^2-i|x-iy|^2$$ While I was expanding the right hand side, I got confused with the $i|x+iy|^2$ part. I know that $ |x+y|^2=|x|^2+2(x,y)+|y|^2$ and ...
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1answer
40 views

Determining if a set is a vector space?

We just started learning vector spaces and I have no idea how to begin solving this problem. I'm asked to determine if the following is a vector space and if it is not I'm then asked to identify at ...
0
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1answer
624 views

Find direction of angle between 2 vectors

I have successfully calculated the magnitude from one vector to another using: $$\cos^{-1} \left(\frac{u \cdot v}{||u||\,||v||}\right).$$ However this does not tell me whether this angle is left or ...