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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42 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
81 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
72 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
255 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
44 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
109 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
327 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
74 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
100 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
69 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
133 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
132 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
155 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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1answer
148 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
39 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
162 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
111 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
436 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
67 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
235 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
63 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
73 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
51 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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73 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
427 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
46 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 ...
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3answers
473 views

Prove subset of $S$ is 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 $V$. So if I let $\beta={u_1, ...
2
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1answer
138 views

Basis for recurrence relation solutions

So, I have a question: Imagine a recurrence relation $U(n+2) = 2U(n+1) + U(n)$. How do I determine the dimension (and the vectors that constitute the basis) of a vector space which contains all ...
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1answer
72 views

functions and inner product spaces

if $l$ is a real valued function defined on $\mathbb R^n$ by $l(x) = \langle x,y\rangle $where $\langle\cdot,\cdot\rangle$ is some inner product on $\mathbb R^n$ and $y$ is a fixed vector in $\mathbb ...
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1answer
82 views

projection onto vector spaces

How do you project a vector on to the euclidean ball? For example, if there is a vector $x ∈ R^n$ how does one project this onto the euclidean ball. What are the steps for projecting a vector onto a ...
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0answers
30 views

$\max_{1 \leq i \leq n}|\langle x,w_i\rangle|$, $\max_{1 \leq i \leq n}|\langle y,w_i\rangle|$ at same $w_i$ if $x$ and $y$ are close enough?

Let $x,y \in \mathbb{C}^n$ with $|x|_2 = |y|_2 = 1$. Let $w_1, \ldots, w_N \in \mathbb{C}^n$. Let $j,k \in \{1,\ldots, N\}$ such that $$ |\langle x,w_j\rangle| = \max_{1 \leq i \leq n}|\langle ...
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2answers
105 views

Is there a linear transformation who domain isn't all of $\mathbb{R}^n$?

My prof said that for a linear transformation: $$T: \mathbb{R}^n \rightarrow \mathbb{R}^m$$ for some real $n$ and $m$, $\mathbb{R}^n$ is called the domain. But some "normal" functions have domains ...
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2answers
6k views

Reflection across a line?

The linear transformation matrix for a reflection across the line $y = mx$ is: $$\frac{1}{1 + m^2}\begin{pmatrix}1-m^2&2m\\2m&m^2-1\end{pmatrix} $$ My professor gave us the formula ...
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0answers
41 views

Indecomposable vector space

Let us define $V$ as a quotient space $\mathbb{K}[t]/(p^m)$, where $p$ is an irreducible polynomial. Condsider the linear operator $\phi\in Hom_{\mathbb{K}}(V,V)$, which sends each $q+(p^m)$ to ...
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1answer
125 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
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2answers
55 views

How to show the vector $V=C_1V_1+C_2V_2+C_3V_3$

Suppose that $V_1,V_2$, and $V_3$ are mutually perpendicular non-zero vectors in 3D and $V$ any vector in 3D. How could I show that $V=C_1V_1 + C_2V_2 + C_3V_3$ where $C_i = (V\cdot V_i)/\lvert ...
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2answers
184 views

Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.

Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed. I'm having trouble in ...
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3answers
88 views

How to find a suitable orthogonal matrix

Assume $x,y \in \text{R}^n$ are two vectors of the same length, how to prove that there is an orthogonal matrix $A$ such that $Ax=y$? Thanks for your help.
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2answers
49 views

How to interpret involutory change of basis transformation?

Just working through an assignment and a change of basis matrix popped up which was involutory - its own inverse. I am not quite sure how to think about this... Presumably it means that the ...
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2answers
54 views

How many non-isomorphic subpaces does $\mathbb{R}^4$ have?

Considering the vector space $\mathbb{R}^4$, I was thinking of the following basis: $\emptyset$, the zero point $(0,0,0,0)$. $\{(1,0,0,0)\}$, $\{(0,1,0,0)\}$, $\{(0,0,1,0)\}$ and $\{(0,0,0,1)\}$. ...
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2answers
105 views

Vector space basis?

Hey can some help me with this practise question I don't even know where to start thanks Let $V$ be a vector space and let $\{v_1, \dots , v_n\}$ be a basis for $V$. Show that $\{v_1,v_1+v_2, \dots ...
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1answer
28 views

Conjugate vectors

What are conjugate vectors? Can I have an example of it? [ This question is in respect to finding the roots of equations with conjugate direction methods]
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1answer
29 views

What direction does a vector with more than two entries point at?

Say you are given theses two vectors: u = (1, -2, 4) v = (-2, 4, 8) Since there are three entries, how do you know if they point in the opposite/same/different direction?
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1answer
167 views

Is the set closed, open, or neither?

Consider $C[0,1]$, the normed linear space of all real-valued continuous functions within the given interval. The norm endowed on this space is $\|f\|_{\infty} = \sup_{x \in [0,1]} |f(x)|$. Consider ...
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2answers
101 views

Can you transform any coordinate from any “space” to another “space” that's defined?

This question pertains to Matrix Transformations. So to provide an example, if I have 3D coordinates where $X = -1$ to $1$, $y = -1$ to $1$, $z = -1$ to $1$. They are "normalized" in my mind. Can I ...
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3answers
484 views

Quotient spaces in linear algebra

There's a statement in some notes I'm reading that goes like this: "...$V/U$ is a 'simplified version' of $V$ where the elements of $U$ are ignored" ($V$ and $U$ are vector spaces). I'm still ...
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1answer
47 views

Matrix of transformation

I have just finished understanding the topic of matrix of the transformation, and we just started T-invariant subspaces. Can anyone please help me with these questions, because I don't know how to ...
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2answers
50 views

Can the intersection of two non subspace subsets be a subspace?

I feel as though it cannot, but I'm having difficulty explaining why.
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3answers
2k views

Writing u as a linear combination of the vectors in S.

Write vector u = $$\left[\begin{array}{ccc|c}2 \\10 \\1\end{array}\right]$$ as a linear combination of the vectors in S. Use elementary row operations on an augmented matrix to find the necessary ...
7
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3answers
477 views

Differences between infinite-dimensional and finite-dimensional vector spaces

I've just started a course in Representation Theory, and in solving our first homework I've used a couple of theorems about finite-dimensional vector spaces (for an example, rank-nullity theorem). My ...