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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2answers
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If $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field , then $T^n$ is the identity operator?

If $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field , then is it true that $T^n = I$ ( the identity operator) for some positive integer $n$ ?
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1answer
32 views

Show that S is a real vector space using the standard operations on R3 (what are “standard operations” on R3?)

My problem is: Let S = {(x,y,0):x,y E R}. Show that S is a real vector space using the standard operations on R3. what exactly are the standard operations on R3? I'm not sure if it means closed ...
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3answers
63 views

Let $V$ be a finite dimensional real vector space and let $A:V\to V$ be a linear map such that $A^2=A$

Let $V$ be a finite-dimensional real vector space and let $A:V\to V$ be a linear map such that $A^2=A$. Assume that $A$ not $0$ or $I$. Then show that $1.$ $\ker(A)$ is not $\{0\}$. $2.$ ...
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1answer
272 views

Problem with alternate solution — Equation of plane through point and containing intersection line of two planes [Stewart P $803, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
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1answer
22 views

Linear algebra: a finite subset that generates a finite-dimensional vector space

Let $V$ be a finite-dimensional vector space and $S\subset V$ a subset that generates $V$. How can I show that there is a finite subset of $S$ that generates $V$?
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2answers
48 views

Some Difficult Questions on Quotient Spaces in Linear Algebra

I've stumbled across some problems in the quotient spaces and I solved many of them but I cannot figure out the following. These are not homework and I appreciate your help. a) Let $P$ be the space ...
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0answers
11 views

Angle between two vectors from specific viewpoint

I have two vectors, $u$ and $v$, which both are non-zero, three-dimensional and starting from origin. I'm viewing the vectors from point $c$. In this case we can assume that $c$ is a vector from the ...
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0answers
21 views

Position Vector - Using Geometry

Given Data and Specifications in the problem $O_i,O_j$ represent two co-ordinate systems with base orthogonal frame vectors $x_j,y_j,z_j$ and another frame $x_i,y_i,z_i$ with respect to that ...
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0answers
43 views

Determining a basis for certain addition and multiplications defined

Let $\mathbb W=(a,b) \in \mathbb R^2 \|b>0$ and define addition by $(a,,b)+(c,d)=(ad+bc,bd) $ and define scalar multiplication by $k * (a,b) =(kab^{k-1},b^k)$. Find a basis for $\mathbb W$ and ...
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1answer
34 views

How to determine if the subset is closed under scalar multiplication and vector addition [duplicate]

How to determine if the following subset is closed under scalar multiplication and addition. The subset W, $$W = \left\{\left[\begin{smallmatrix}a\\b\\c\end{smallmatrix}\right]: 3a - 5c = 0\right\}$$ ...
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4answers
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Why are vector spaces not isomorphic to their duals?

Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$). I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the ...
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1answer
22 views

how to obtain transformation matrix A in y = Ax + b notation?

I'm trying to obtain original transform matrix A and its translation vector b From y=Ax+b equation. I have original values of vectors before transform and translation (x) and vectors after transform ...
4
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1answer
103 views

Let $a_1, …,a_n , b_1,…b_n$ be $2n$ distinct elements of a field , then is the matrix $\Big(\dfrac1{a_i-b_j}\Big)_{ij}$ non-singular?

Let $a_1, ...,a_n , b_1,...b_n$ be $2n$ distinct elements of a field and define $$h_{ij}:=\dfrac1{a_i-b_j} , \forall i,j=1,2 ,\cdots,n $$ then is the $n \times n$ matrix $H:=(h_{ij})$ non-singular ? ...
2
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0answers
12 views

Set of polynomials a subspace of P3?

Is the set of all polynomials of the form a0+a1x, where a0 and a1 are real numbers a subspace of P3? My book says it is not. Both closure under addition and scalar multiplication hold, so I don't ...
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0answers
21 views

Set of vectors form a subspace of R3

Do all vectors of the form (a,b,c),where b=a+c, constitute a subspace of R3? I checked for closure under addition and scalar multiplication and found them both to hold, but my book answer key says ...
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1answer
26 views

Line integrals of a vector field

I had no problem with part (a) or (d). The sketch was fine, and if we were given the F in part (d), we can use Stokes' Theorem. However, I am struggling with (b) and (c). For (b) I know I must ...
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1answer
33 views

Matrix/Vector Calculus

I studied civil engineering almost 20 years ago and forget some knowledge of maths. Hope I could get some help here... Here are some pictures: ...
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1answer
96 views

For $n \ge 2$ , does every linear operator on $\mathbb R^n$ has an invariant subspace of dimension $2$ ?

Is it true that for $n \ge 2$ , every linear operator $T$ on $\mathbb R^n$ has an invariant subspace of dimension $2$ ? I know that $T$ always either have a $1$ or $2$ dimensional invariant subspace ; ...
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1answer
34 views

Can anyone explain what this transformation means?

I don't understand T: V->E where V is a subspace of Euclidean space What is the difference between Euclidean space and linear space?
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0answers
15 views

Different SVD results in Matlab

my question relates to calculating SVD in Matlab. I have been reading a lot and somehow I have jumbled up all the facts. It would be great if you experts could get me to the right track. My task is ...
2
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0answers
27 views

Linear Operator and Matrices [duplicate]

So, this is question that I got for my homework. If anyone could explain the question and tell me the solution, I would be really happy. Thanks. :) Given $V$ be a 2-dimensional real vector space with ...
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2answers
36 views

What is the special name of vectors like <0,1,0,0,0> or <1,0,0>?

Is there a special name for vectors whose elements are all 0's except one 1? Thanks.
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1answer
16 views

Nonzero function of zero norm in Hilbert space

Does the function, $$f(x)=\left\{ \begin{matrix}1 \,\,\,\,\,\,\,\,\,\,\,\text{ if }x=0\\ 0 \,\,\,\,\,\,\,\,\,\ \text{ if }x\ne0 \end{matrix} \right.$$ belong to $L^p$? If yes, how do we reconcile ...
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0answers
28 views

General Steinitz exchange lemma

Where can I find a proof of the following general Steinitz exchange lemma: Let $B$ a basis of a vector space $V$, $L\subset V$ be linearly indepdent. Then there is an injection $j:L\rightarrow B$ ...
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votes
4answers
2k views

*understanding* covariance vs. contravariance & raising / lowering

There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors - after struggling through many of them, I think I'm starting to get the idea. ...
5
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2answers
45 views

If some vectors in $\mathbb Q^n$ are linearly independent over $\mathbb Q$ , then are they also linearly independent over $\mathbb C$?

Let $\vec v_1 , ..., \vec v_k $ be vectors in $\mathbb Q^n$ linearly independent over $\mathbb Q$ , then is it true that $\sum_{i=1}^ka_i\vec v_i=0, a_i\in \mathbb C, \forall 1\leq i\leq k \implies ...
0
votes
1answer
11 views

Bounded Linear Maps on Normed Vector Spaces

Let $A$ be an $m\times n$ matrix $(\alpha_{jk};\;j=1,...m,k=1,...,n).$ As we know, $$[Bx]_j = \sum_{k=1}^n\alpha_{jk}x_k,\;\;\;\;\;j=1,...,m,\;\;\;x=(x_1,...,x_n),$$ defines a bounded linear operator ...
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0answers
17 views

Orthogonal projection of Parabola over straight line.

I am wondering how to find the orthogonal projection of one real function over another. I do understand the orthogonal projection of a vector in finite dimensional vector space but don't know how to ...
2
votes
3answers
77 views

Linear Algebra: What do vector spaces represent?

I understand what a vector can represent, but I still don't understand what a vector space represents. I understand that you can add two vectors and that becomes a vector space. What else can you do ...
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1answer
48 views

what does “closed subspace” in papers mean?

In many books and articles one finds sentences like this: "let $A$ be a closed subspace of ...". Now my question might be stupid, but I am always wondering what they mean by closed subspace? Is this ...
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0answers
32 views

$x \cdot x$ in inner product space is a quadratic form

Given an inner product space with some inner product $\cdot$ , how can I prove that $x \cdot x$ for any vector $x= (x_1,... x_n)$ is a quadratic form in $x_i$? I know how to recover an inner product ...
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3answers
116 views

$I+A^*A$ is non-singular whenever $A$ is a square matrix with complex entries? [closed]

Let $A$ be a square matrix with complex entries , then is it true that $I+A^*A$ is non-singular ? where $A^*$ denotes the conjugate transpose of $A$ http://en.wikipedia.org/wiki/Conjugate_transpose ...
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1answer
45 views

Prove that B is vector subspace [closed]

In $ℝ^∞$ we've got B={$\{x_n\}^{\infty}_{n=0}| \exists K\in~ (0,\infty) ∀n\in \mathbf{N}, | x_n|\leq K n^{-1}$}. Prove that B is vector subspace.
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1answer
37 views

Vector subspace of polynomials

If I have a set of polynomials of degree at most $2$, such that $p(x) \geq 0$ for any real $x$. It isn't a vector subspace because I can multiply by a negative number such that $p(x) < 0$?
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2answers
18 views

Determining whether sets of vectors form a basis

Is $\{(1,1,0,0),(0,0,1,1)\}$ a basis for the subspace of $\mathbb{R}^4$ consisting of all vectors of the form $(a,a+b,b,b)$ with $a,b\in \mathbb{R}$? Here is how I proceeded: First note that ...
0
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0answers
16 views

Does the operation of addition on the subspaces of V have an additive identity? Which subspaces have additive inverses?

I was reading Linear Algebra Done Right. I came across the following question (Ch-1, Q12), for which I have solution , but I am having little confusion regarding it: Q12. (a) Does the operation of ...
2
votes
1answer
47 views

Does all Eigenvectors of $A$ lie on the vector space of $Ax$?

The problem is with the last part of the following question: I will write my results to the first parts which are correct here : Three Eigenvalues: $$\lambda_1=1 , \lambda_2=2 , ...
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votes
5answers
281 views

How to solve this to find the Null Space

What I did : I put this into reduced row echleon form: $$\begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 ...
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votes
2answers
33 views

Are all vector spaces also a subspace?

I am currently learning about vector spaces and have a slight confusion. So I know that a vector space is a set of objects that are defined by addition and multiplication by scalar, and also a list ...
0
votes
1answer
28 views

Need help to understand a line of a proof of diagonalizability of real symmetric matrices

I was reading a proof of diagonalizability of real symmetric matrices using the concept of generalized eigenvalues and understood all except the very starting (and fundamental) line of the proof " if ...
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votes
1answer
13 views

can't get a solution to a vectors question

We have : $|\vec{u}|=1 ,|\vec{v}|=5 ,\vec{u}\cdot\vec{v}=3$ $\vec{w}=(\vec{u}-\vec{v})\times(2\vec{u}+\vec{v})$ we have to calculate $|\vec{w}|$ after playing with equations we have in our ...
1
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2answers
60 views

Let ${v_1,v_2,…,v_n}$ be a linearly independent subset of a vector space V where $n \geq 4.$ [closed]

Let ${v_1,v_2,...,v_n}$ be a linearly independent subset of a vector space $ V $where $n \geq 4.$ Set $w_{ij}=v_i-v_j.$ Let $ W $ be the span of $\{w_ {ij} : 1\leq i,j \leq n\}$. Then which is correct ...
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0answers
32 views

Need some help calculating vectors.

What I am attempting to do is use some math that is frankly beyond me to draw a line in three dimensional space. I am using vector subtraction to create an imaginary line from point A to point B ...
0
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1answer
35 views

Is this a valid operator norm?

The "norm" (yet to be proved or disproved) defined for a matrix $A \in \mathbb{C}^{m\times n}$ by $\|A\|=\max_{i,j}|A_{i,j}|$. Is $\|\cdot\|$ a valid operator norm? (I think it is. As it satisfies ...
0
votes
0answers
21 views

Product of dot products of two vectors

I have a product of innerproduct/dot product of two vectors. $ \langle u_i,v_j \rangle\cdot\langle x_i,y_j\rangle$. Is there any transformation/decomposition such that I can combine $u_i$ with $x_i$ ...
0
votes
1answer
22 views

How to show this iff relationship?

Let $M$ be a linear operator in $L(V)$, where $V$ is a vector space. How to show $\ker(M^k) = \ker(M^{k+1})$ iff $R(M^k) \cap \ker(M) = \{0\}$. It is not difficult to show $\ker(M^k) \subset ...
2
votes
1answer
15 views

Proving that a set of vectors is a basis in P_3

I want to show that the following set constitutes a basis for the vectorspace of polynomials up to degree 3, i.e. $P_3$: {(t-1),(t+1),(t-1)^3,(t+1)^3)} Since $P_3$ is four dimensional, I believe it ...
0
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0answers
21 views

volume parallelepiped-linear algebra

So I have this exercise where they give me the vertices and i must chose those to use to calculate the volume= absolute value of the transformation matrix. The matrix will be $3\times 3$, so i am ...
2
votes
3answers
53 views

What is the mathematical meaning of $xx^T$

Assume x is a n by 1 column vector, then , it is easily known that $x^Tx$ is the sum of squares. When calculate $xx^T$, it will give you a n by n matrix. I am wondering what is the mathematical ...
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0answers
19 views

Discrete Geometry (Polytopes)

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset R^d$ be a point configuration affinely spanning $R^d$ (i.e., $aff(V) = R^d)$. Let H be the collection of hyperplanes spanned by ...