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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36 views

If $a=b+ c$ and $E$ is the basis of $a$, Will $E$ be also basis for $b$ and $c$?

Suppose $a$ lies in the span of a set of independent vectors $E$. Now, if $a=b+c$, is it also the case that $b$ and $c$ lie in the span o the same set of vectors $E$? if the question is obscure, ...
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1answer
160 views

Clarification: Viewing $\mathbb{R}^n$ as a probabilistic state space

In this MathOverflow post on visualizing high-dimensional spaces, Terry Tao states that "the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be ...
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3answers
143 views

$T: V\rightarrow W$ is an injective linear transformation when restricted to subspace $A$ of $V$. Then can we conclude that $\dim(A) = \dim T(A)$

Let $V$ and $W$ be finite dimensional vector spaces and let $A$ be a given subspace of $V$: Also, we have a linear transformation $T: V\rightarrow W$ such that $T$ is injective on the subspace $A$. ...
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2answers
121 views

How to tell if a point is between two others in $\mathbb{R}^3$

Find the coordinates $c$ of the point $C$ $st$ $C$ is on the line from $A (2, -3, 1)$ to $B (8, 9, -5)$, it is between A and B, and $\vec{AC} = 2\vec{CB}$. So, I found the vector $\vec{AB}$ ...
2
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1answer
578 views

Direct sum of subspaces

I am rather confused. Suppose $V$ is a finite dimensional vector space and $A,B,C$ are (non-trivial) subspaces of $V$ such that $V=A\oplus B=A\oplus C=B\oplus C$ and it is said that there is a ...
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1answer
73 views

a certain set of polynomials forming a subspace

Just out of curiosity: Why is it that all polynomials vanishing on an irreducible component in some affine space form a vector subspace? Thanks for your time.
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1answer
110 views

Vector question, solving $r\wedge a=b$ and $r\wedge c=d$, with conditions

I am stuck on the following Show that the vector equation $r\wedge a=b$ has a solution $$r=\lambda a + \frac {a \wedge b}{|a|^{2}}$$ Show that the vector $r\wedge a=b$ and $r\wedge c=d$, with ...
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1answer
511 views

Difference between Dimension of a Linear transformation (space) and the Dimension of its Column Space?

As my title suggests, what exactly is the difference? The Column Space of a transformation $T: \mathbb R^n \to \mathbb R^m$ is simply the subspace which "contains" all the possible Images, right? If ...
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1answer
120 views

Intuitive interpretation of the 3D to 2D mapping

Suppose a 3D configuration of points is given, $X\in\mathbb{R}^{n\times 3}$, and a non-zero matrix $Q\in\mathbb{3\times 2}$, with orthonormal columns. Now, suppose a mapping to 2D is obtained as ...
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2answers
359 views

Norm-preserving map is linear

How can one show that a norm-preserving map $T: X \rightarrow X'$ where $X,X'$ are vector spaces and $T(0) = 0$ is linear? Thanks in advance.
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3answers
82 views

Show that the dimension of a particular linear space is $2$

Question: A Linear transformation $T: \mathbb R^4 \to \mathbb R^4$ is represented by the matrix $$\mathbf A=\begin{pmatrix} \\1&-1&2&3 \\ 2 & -3 & 4 & 5\\ 5 & -6 & ...
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1answer
119 views

V is a vector space such that $V = A\oplus A^\perp$ also $V = A \oplus C$ then can we say that $A^\perp = C$?

I have a vector space $V$ such that $V = A\oplus A^\perp$ i.e. $V$ is a direct sum of its subspace $A$ and orthogonal complement of $A$. Suppose we also have $V = A \oplus C$ Then can we say that ...
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1answer
2k views

Finding the norm of vectors

When finding the norm of the vector: Find $\|2w-2y\|$ such that $w=(1/2,3,1)$ and $y=(0,-1,3/2)$. answer: $$\begin{align*} &2(1/2,3,1)= (1,6,2)\\ &2(0,-1,3/2) =(0,-2,3)\\ ...
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1answer
117 views

Reducing the span of vectors

Out of interest what would be the best way to describe the spanning set of vectors a a and b a=(0,3,-2) b=(1,0,0). Apart form a and b, what other vector belongs to the spanning set? Do i have to ...
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2answers
85 views

Finding the dot product.

Finding the dot product of $(-2w) \cdot w$ where $w=(0,-2,-2)$ $$ \text{dot product} = \frac{v . u}{\| v \| . \| u \|} $$ So $$-2 (0,-2,-2)=(0,4,4) \\ (0+4+4) = 8 \\ (0,-2,-2)=-4 \\ 8 \times ...
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1answer
88 views

Orthogonal vectors question

Can orthogonal vectors have some values that are the same? such as are (1,2,5) and (1,2,-5). orthogonal as the dot product is zero? Thanks in advance!
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1answer
353 views

Symmetric power of vector space

Let $V$ be a vector space over a field $k$ of char. zero and denote by $Sym^n_k V$ its $n$-th symmetric power over $k$. Now I simply want to know what $Hom_k(V,Sym^n_k V)$ is for $n \geq 2$. To be ...
3
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2answers
169 views

$2\times 2 $ matrices over $\mathbb{C}$ that satisfy $A^3=A$

Let $A$ be a $2\times 2$ matrix with complex entries. What would be the number of $2\times 2$ matrices $A$ that satisfies $A^{3} = A$. Question was are they infinite? If it is $3\times 3$ matrix then ...
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1answer
99 views

nilpotent - derivative - why is characteristic important?

Let $V$ be the space of all $f(t) \in K[t]$ with $\mathrm{deg} f \leq n-1$ and let $\psi: V \to V$ with $\psi(f) = f'$. Further $\mathrm{char}(K) = 0$. Then $\psi$ is nilpotent. Since one can take ...
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1answer
184 views

Is there a name for this $k$-fold vector product?

Let $V$ be a set of vectors of length $n$. Define a $k$-fold product on $V$, $$ \Upsilon(\{v_1,\ldots,v_k\}):=\sum_{j=1}^n\prod_{i=1}^k v_{ij}, $$ where $v_i\in V$ and $v_{ij}$ is the $j^\text{th}$ ...
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1answer
83 views

Angle of 2 vectors

Suppose I have 2 vectors $\vec{a}= (5,1)$ and $\vec{b}= (2,4)$. I want to compute the angle between them. See my calculations below. Supposedly the answer is $35.18^{\circ}$ degree but my answer as ...
0
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3answers
99 views

Prove that a space is a subspace

I have 2 subspaces of $M_2({\bf R})$: $U =\left\{ \pmatrix{a&b\cr c&d\cr} : c \ge 0 \right\}$ $V = \left\{ \pmatrix{a&b\cr c&d\cr} : c + 2d = 0, a + b - 2c = 0 \right\}$ I need to ...
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3answers
306 views

What does an inverse matrix abstracts?

I am trying to understand inverse matrixes more in depth. I took the simplest example: 2 points in a 2d space and put it into a matrix. $$\begin{pmatrix}5&7\\-2&3\end{pmatrix}$$ ...
1
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1answer
25 views

Determine matrix of a set in a certain base

I have a set $S = \{ x^2 + 1, x + 1, 1 - x, x^3 \}$ in a polynomial vector space. How do I write a vector matrix of $S$ in the base $B = \{ 1, x, x^2, x^3 \}$? I attempted this using the formula: ...
2
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1answer
324 views

Subspace of V is also a null space of T

Prove that any subspace of vector space $V$ is a null space over some linear transformation $V \rightarrow V$. So far I have: Let $W$ be the subspace of $V$, let $(e_1, e_2, \ldots, e_r)$ be the ...
2
votes
1answer
119 views

Find two vectors that don't belong to a vector space

I have a vector space $L$, which is a subspace of $\mathbb{R}^4$, spanned by these vectors: $$(4, 1, 1, 2), (2, 3, 1, 0), (-10, 35, 5, -20), (2, 13, 3, -4).$$ I need to find two vectors from ...
15
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1answer
811 views

Cardinality of a Hamel basis

What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...
0
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1answer
40 views

Path contained in Surfaces

$y(t)$ is a path contained in two surfaces: $x^2+y^4+z^6=3$, $x+y^2=y+z^2$ also $y(0)=(1,1,1)$ and $||y'(0)||=1$ Need to find the vectors $-y'(0)$ and $+y'(0)$ To be honest, I'm not sure how to ...
0
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1answer
221 views

Determine linear operator image?

I have a simple linear operator: $$\begin{align}g: \Bbb{R^4} &\to \Bbb{R^3}\\g (x, y, u , v) &= ( x + u, x + v, y + u)\end{align}$$ How would I determine the image of this linear ...
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1answer
368 views

The number of subspaces of a vector space

Let $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$, and let $U$ be a subspace of dimension $k$. I want to compute the number of subspaces $W$ of $V$ of dimension $m$ such that $W\cap U=0$. ...
0
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2answers
43 views

Divergence operator and integrals

When is it alright to put the divergence operator into an integral? For example, would the following be right: $$\nabla\cdot\int {1\over |\vec{r}-\vec{u}|}d\vec{u}=\int \left(\nabla\cdot{1\over ...
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1answer
1k views

Parametric equation for a line which lies on a plane

Struggling to begin answering the following question: Let $L$ be the line given by $x = 3-t, y= 2+t, z = -4+2t$. $L$ intersects the plane $3x-2y+z=1$ at the point $P = (3,2,-4)$. Find parametric ...
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4answers
854 views

Easiest way to prove that an operator is linear?

I have an operator $$h: \mathbb R^4 \to \mathbb R^3\text{ given by } h(x, y, u, v) = (2x + 3y - u + 2v, x - 5y + 6v, 2y + u + v)$$ What is the easiest way to proove that this operator is linear? I ...
0
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2answers
108 views

Determining the dimension and a basis for a vector space

I have the following problem: Let $W$ be a vector space of all solutions to these homogenous equations: $$\begin{matrix} x &+& 2y &+& 2z &-& s &+& 3t ...
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3answers
2k views

Question about basis and finite dimensional vector space

I have seen the statement "Every finite dimensional vector space has a basis." (Here on page 5) I'm confused about what this tells me. It seems to tell me nothing: by definition, the dimension of a ...
2
votes
1answer
255 views

Calculating angle of human joint beyond 180° in 3D

I'm having some trouble calculating the angle of an human joint in 3D using the Microsoft Kinect. Here's an example of the angle of the elbow (using the shoulder and wrist joint): Image of example ...
3
votes
1answer
480 views

For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.

I want to prove that for Banach space V there is a compact topological space $X$ so that $V$ is isometrically isomorphic to a closed subspace of $C(X)$-continuous function on a (compact) topological ...
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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 ...
2
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1answer
116 views

find new base vectors based on old ones and the given matrices in both bases

I've got the following matrix $A$ for an endorphism within a base $v_1, v_2, v_3$ $$ A = \left( \begin{array}{ccc} 0 & 0 & -1 \\ 1 & 0 & -3 \\ 0 & 1 & -3 \\ \end{array} ...
2
votes
1answer
288 views

Finding a bilinear map that differs on two given points.

Let $V,W$ be finite dimensional real vector spaces, and let $(v,w) \not= (x,y)$ be two points in $V \times W$. Is it possible to construct a bilinear map $\alpha: V \times W \to \mathbb{R}$ such that ...
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1answer
190 views

Column Space of Square Matrix

Assume that I have a $3 \times 3$ matrix $A$ with columns $A_1$, $A_2$, and $A_3$ that are linearly independent. Say that I want to find the column space of A. Isn't it possible for me to find some ...
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1answer
113 views

basic vector being hermitian

If the space has a mixed metric signature, not all the basis vectors are Hermitian. Nevertheless, they are defined to be self-adjoint under reversion. The vector transpose conjugate is, ...
7
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1answer
475 views

Showing that the dual space of bilinear maps $V \times W \to \mathbb{R}$ satisfies the tensor product property, for finite dimensional vector spaces.

Let $U,V$ and $W$ be finite dimensional vector spaces, and define $B$ to be the vector space of all bilinear maps $V \times W \to \mathbb{R}$. Given a bilinear map $\alpha : V \times W \rightarrow U$, ...
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2answers
386 views

Vectors / Linearly independent question

an easy question that I completely understand, just not sure how to algebraically prove. $u,v,w$ are vectors in $R^3$ given $u\times v +v\times w + w\times u=0$ I need to prove that {u,v,w} are ...
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2answers
156 views

Vectors question

I'm trying to prove whether the followings statements are true or not. I would appreciate your help, as I'm not sure how to begin. Given: $ u,x_n \in \mathbb{R}^3$ and for every $n$, let $x_{n+1}=u ...
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1answer
248 views

The range of dot product of two vectors under certain constrains.

I am thinking of the maximal value and minimal value of two vectors in some special case. Following is my problem statement. $\max\sum_{i=1}^Nx_iy_i$ and $\min\sum_{i=1}^Nx_iy_i$ subject to the ...
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1answer
49 views

Dimension of a set and its closure are equal in an Inner-product space?

I want to show that, given a subset $M$ of an Inner Product space $X$. If $M$ is a total set then, $M^\perp=\{0\}$. Which I have shown using the completion of $X$, which will be a Hilbert Space. And ...
5
votes
4answers
1k views

Dot product of two vectors without a common origin

Given two unit vectors $v_1, v_2\in R^n$, their dot product is defined as $v_1^Tv_2=\|v_1\|\cdot\|v_2\|\cdot\cos(\alpha)=\cos(\alpha)$. Now, suppose the vectors are in a relation $v_2=v_1+a\cdot1_n$, ...
0
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1answer
99 views

Questions regarding the column and row spaces of Echelon form

I have several conceptual questions that have been confusing me for a while in linear algebra. Let A be a 3 by 5 matrix with full rank rows. A is now simplified to Echelon form U and further ...
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1answer
58 views

Find matrix $A$ of $f$ with respect to the standard bases of $V$ and $W$?

Suppose $V=\Bbb{R}^3$ and $W=\Bbb{R}^2$. Let $f:V\to W$ such that $f(x,y,z)=(zx+z,3y)$. Find matrix $A$ of $f$ with respect to the standard bases of $V$ and $W$?