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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1answer
28 views

Can we construct two sets and functions for the given conditions?

Can we construct two sets $A$, $B$ and two invertible functions (one to one) $f_A \in \mathbb{R}^n$, $f_B\in \mathbb{R}^n$ such that the following conditions are satisfied? The conditions are ...
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2answers
51 views

Easiest way to find a vector in a span

So $V=\text{span}\{v_1, v_2\}$ where $v_1 = (1, 1, -1)$ and $v_2 = (1, -1, 2)$ I have been given four different vectors and must find which vectors are in $V$. What is the most efficient way to do ...
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0answers
46 views

Conditions such that a given set $X$ qualifies as a complex vector subspace

I am struggling with the following problem as introduced in Jänich's Linear Algebra. $$\mathbb{C}^2:= \lbrace (z,w) \mid z,w \in \mathbb{C} \rbrace $$ Which conditions have to be applied to $a,b \in ...
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2answers
34 views

map colinear triple of points to another triple of points in $\mathbb{R}^2$

Given two triples of pw different colinear points in $\mathbb{R}^2$ so $(x_1,x_2,x_3),(y_1,y_2,y_3) \in (\mathbb{R}^2)^3$. There is a map of the form $T:\mathbb{R}^2\to\mathbb{R}^2,x\mapsto Ax+b$, ...
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0answers
55 views

Vector calculus: Finding a model fit from a 2D grid

First of all, I am a biologist. My knowledge of math is very limited. Therefore, I come here for help, but please take into consideration that I may not be very good at asking the question in a good ...
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1answer
26 views

Subspaces of $\mathbb(R)^3$

Given $U=\{(1, -1, 3)^t\}$ and $V_a=\{(x, 3x-az,z)^t\}$ for any $a\in\mathbb{R}$, how to determine $a$, such that $U \cap V_a = \{o\} \land U+V_a=\mathbb{R}^3$? I already found out that $V_a$ always ...
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1answer
36 views

Proving mutual orthogonality of vectors

Let three vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ in three-space be such that: $$ a_ia_j + b_ib_j + c_ic_j = \delta_{ij} $$ where a vector name with subscript represents a component of ...
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1answer
37 views

Hyperplane multiplication and sum

Lets assume that I have a hyperplane defined by vectors: w.x + b => is the equation of the hyperplane, w,x,b are vectors what happens to the hyperplane if I multiply it by a constant? for eg: 7*w.x ...
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1answer
44 views

Does this sequence span $V$?

Let $F$ be a field. $V$ is a vector space over $F$, consisting of all polynomias of degree less or equal to $3$ with coefficients in $F$. Does the sequence $S:\,x,(1-x^2),x^3$ span $V$ ?
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1answer
32 views

What is the $\lVert v\rVert$ sign mean in the context of vectors?

Suppose $V$ a inner product space, $u, v \in V$. I need to prove this identity: $$\lVert u+v\rVert^2 +\lVert u-v\rVert^2 = 2\left(\lVert u\rVert^2 +\lVert v\rVert^2\right) $$ what is the $\lVert ...
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1answer
32 views

Orthonormal Sets and the Gram-Schmidt Procedure

What my problem in understanding in the above procedure is , how they constructed the successive vectors by substracting? Can you elaborate please?
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3answers
239 views

Understanding Span, Basis, and Dimension

I am a bit confused with span, basis, and dimension (when dealing with vector spaces). My teacher told us that a span is a finite linear combination. And I know that a basis is a spanning, linearly ...
2
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1answer
63 views

Why and how are quaternions 'bilinear'?

What does it mean when we say that quaternion composition is 'bilinear'? I have observed that some authors write quaternion multiplication as: While others specify: Excuse the poor images, ...
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4answers
106 views

Why is the inner product not an element of the Hilbert space?

What I know about Hilbert space is that, elements in that space can be complex numbers. But I was confused to read this statement from a book: The inner product, being a complex number, is not an ...
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2answers
123 views

Vector Subspaces- Counterexample

ive been struggling to come up with a counter-example, ive been treating V as R^3. I would very much appreciate if someone could come up with a counter example and the right equality involving the ...
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1answer
103 views

General Linear Group of a vector space

I am reading a text on Lie groups. There is a whole chapter devoted to the group of invertible real or complex matrices of degree n, which are called the General Linear groups(complex and real). ...
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1answer
79 views

How to prove $a_1^Ta_1+a_2^Ta_2\le b_1^Tb_1+b_2^Tb_2.$

Let $p,q > 0$, $a_1, b_1\in \mathbb{R}^m, a_2,b_2\in\mathbb{R}^n$ be vectors. Given that ...
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2answers
42 views

Prove that if $T_2\circ T_1$ is one to one, $T_1$ is also one to one.

Suppose that $T_1 : V \to U$ and $T_2 : U \to W$ are linear transformations of vector spaces. If $T_2 \circ T_1$ is one-to-one, prove that $T_1$ is also one-to-one.
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1answer
48 views

Isomorphism between $\mathbb{F}^{m \times n}$ and $\mathcal{L}(V, W)$

Let $\mathbb{F}^{m \times n}$ be the vector space of all $m \times n$ matrices and let $\mathcal{L}(V, W)$ be the vector space of all linear maps from a vector space $V$ to a vector space $W$ ($V$ and ...
2
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1answer
112 views

Show that the set of all functions described by $y(t) = c_1\cos\omega t + c_2\sin\omega t$ is a vector space

Show that the set of all functions described by $y(t) = c_1\cos\omega t + c_2\sin\omega t$ is a vector space. ($c_1,c_2$ are constants and $\omega$ is fixed). I am having real issues imagining ...
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1answer
58 views

$E$ an $n$-dimensional vector space. Find all endomorphisms $f$ of $E$ which satisfy $f\circ f = \operatorname{Id}_E$.

Let $E$ be a vector space of dimension $n$. Find all endomorphisms $f$ of $E$ which satisfy $f\circ f = \operatorname{Id}_E$. Is trivial that $f = \operatorname{Id}_E$ is a solution, but I don't ...
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1answer
64 views

Norm, Euclidean Space and Distance

To a complete layman, how would you define the following terms intuitively? $norm$ , $euclidean$ $space$ , and $euclidean$ $distance$ ? Note: I have tagged Linear Algebra and Probability Theory ...
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2answers
85 views

Prove that it's a subspace of $\mathbb{R}^3$

I have the following definition of $V_a$: $$V_a := \{(x, y, z)^T \in \mathbb{R}^3 : y = 3x - az\}, \quad \text{for $a \in \mathbb{R}$}.$$ My first problem: I don't understand this definition. Which ...
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2answers
41 views

What is that sign in the context of vectors?

Suppose $v = (0, -5, 5, -6, -7)$ a vector. I need to find $$\|v\|_1, \|v\|_2, \|v\|_9, \|v\|_\infty.$$ can you please explain me what does $\|v\|_i$ mean?
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0answers
67 views

How to condense a matrix to a vector

I'm not an experienced person in mathematics and this might either sound like a trivial question or a stupid one. However, this problem arose to me when I was writing a program. Following is my ...
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2answers
68 views

Dimension of $R$ over $Z_p$

What is the dimension of vector space of $R$ over $Z_p$ ? I think it is $p$. Fruitful suggestion on how to look at it would be great.
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3answers
233 views

A Vector Space is a Set - Axiom or Derivation?

I understand that structures with the properties of the real and complex numbers can be defined and derived from the axioms of ZFC set theory. But can a structure with the properties of a (possibly ...
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1answer
42 views

linear subsets of linear space

Need help with graduate level work. I've been out of undergrad for almost 20 years. The problem says: Given $B$, a subset of a linear space $L$, $q$ element of $L$ and $p$ a point of $B$ such that ...
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4answers
106 views

Inner Product Space on linear transformation on itself

So $V$ is an inner product space and $T : V \to V$ is a linear map such that $$||T(v)|| = ||v||$$ for all $v \in V$. Prove that $$\langle T(v), T(w)\rangle = \langle v, w\rangle$$ for all $v,w \in V$. ...
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2answers
246 views

Nonzero subspace that is invariant under any operator cannot be proper?

This is not for homework, and I would just a like a hint please. The question asks Prove or give a counterexample: If $U$ is a subspace of $V$ that is invariant under every operator on $V$, then ...
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1answer
54 views

Vectors In Implicit Form

Say I have two vectors in the x,y,z plane in implicit form, where i,j,k are the basis vectors. How do I find out the angle between the two lines? I was considering expressing the two vectors in ...
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1answer
93 views

Find where $r(t)=<t,t,t^2>$ hits the $x-y$ plane

I have to find $r'(t)$ and $||r'(t)||$ for $r(t)=<t,t,t^2>$, which I know how to do. $r'(t)=<1,1,2t>$ $||r'(t)||=\sqrt{2+4t^2}$ The problem is that my professor didn't explain how to ...
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1answer
54 views

Subspaces and span?

Let $S$ be the subspace spanned by $(\text{u}_1, \text{u}_2, ... , \text{u}_m)$. Then, $S$ is the smallest subspace containing $(\text{u}_1, \text{u}_2, ... , \text{u}_m)$ in the sense that if ...
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1answer
112 views

Find basis to quotient space of 2 spaces

My question is as follows: V is the space of all n by n matrices. W is a subset of V, and is defined by the space of all symmetric n by n matrices. We are asked to find a basis for V/W I don't know ...
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1answer
114 views

Signed angle between 2 vectors?

http://stackoverflow.com/questions/2150050/finding-signed-angle-between-vectors on this link I found the following formula: ...
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3answers
129 views

geometric description of subsets that form bases in $\mathbb R^2$ and $\mathbb R^3$

How do you give a geometric description of the subsets of $B$ in $\mathbb R^2$ and $\mathbb R^3$ that form bases? would the bases of the space have the same number of points. I'm having difficulty ...
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0answers
26 views

Find vector with simple trigonometry

I've spent too much time solving a fluid mechanics problem because of this trigonometry. How do I find $V_{t2}$ ? The answer is $rw - V_{n2} \cot(\theta)$. (can't seem to get equations to work)
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1answer
121 views

Extension of a linear map to a commutative graded algebra

Let's fix the notation, $V=\bigoplus_{i\geq 0}{V^i}$ is a graded vector space and $\Lambda V$ is the free commutative graded algebra on $V$. I have been struggling to understand this example: ...
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1answer
36 views

Vector field of functions equalling real numbers

We consider the set $\mathbb{N}^{\mathbb{R}}$ (i.e. all functions $f\colon\mathbb{N}\to\mathbb{R}$). I've been asked to prove that this forms a real vector space (that is, a vector space over ...
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1answer
148 views

Could we write Fourier transform as a matrix?

I have heard that Fourier transform is a linear transformation. I have also heard that any linear transformation can be written as a matrix multiplication. (probably I'm missing some details in the ...
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1answer
100 views

Determining if a homomorphism is an isomorphism

Let $T \in \mathcal{L}(V)$, where $\mathcal{L}(V)$ is the set of linear operators mapping a vector space $V$ to itself, and let $U$ be an isomorphism from $V$ to another vector space $W$. We claim ...
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2answers
304 views

If the row-reduced form of matrix $A$ has a row of zeros, its columns do not span $\mathbb{R}^n$

Can someone explain why it is that, if the row-reduced form of an $n\times m$ matrix $A$ has a row of zeros, the columns of matrix $A$ do not span $\mathbb R^n$? I'm not seeing the bigger picture ...
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1answer
66 views

Finding maximal number of bad triplets

Let $a,b,c\in \mathbb{F}_{3^n}$. The summation of two vectors is done with modulo $3$. The elements of vectors are $0,1$ or $2$. We will say that $a,b,c$ form a bad triplet if $a\neq b,a\neq c,b\neq ...
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1answer
189 views

Find dimension of even polynomials

Let $V$ be a the vector space over $\mathbb R$ of all polynomials with real coefficients. Let $W$ be the subset of all polynomials with only even powers in their expression. So $p(X) \in W$ means ...
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3answers
60 views

Is it possible to be given three points and NOT be able to determine the eqn of the plane through them?

I think that it's always possible but I can't explain my answer. Can someone please help me with this. I may even be wrong (maybe it is possible for there not to be a plane) Thanks in advance!
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0answers
65 views

Orthogonal set summation proof

Let $\{v_1,v_2,....,v_k\}$ be an orthogonal set in $V$, and let $\{a_1,a_2,\ldots,a_k\}$ be scalars. Prove that $$ \left\|\sum_{i=1}^k a_iv_i\right\|^2 = \sum_{i=1}^k |a_i^2| \, \|v_i\|^2. $$ I ...
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1answer
49 views

In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$

In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$, let $W$ = { f ∈ $C(\mathbb{R})$ | $∫_{-∞}^∞$ | f(x) | $dx$ < ∞} where the ...
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0answers
66 views

Extension of multiplication to the tensor algebra.

In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction We construct $T(V)$ as the direct sum of vector spaces $T^kV$ for $k=0,1,2,…$ $$ T(V)= ...
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1answer
351 views

What are non-orthogonal eigenvectors?

Given a symmetric matrix $A$, the maximum of the trace, $Tr(Z^TAZ)$ under the assumption that $Z^TZ=I$ occurs when $Z$ has the eigenvectors of $A$, as $Tr(U^TAU)= \lambda_1 +\lambda_2+...\lambda_ d$ ...
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1answer
45 views

Do the paths intersects? If so where

There are two unidentified objects in the sky. The path of the first object is given by $r(t)=\langle t,-t,1-t\rangle $ and the second object's path is $s(t)= \langle t-3,2t,4t\rangle$ Do the paths ...