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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$\mathbb R^2$ as a plane

What elements allow me to say that $\mathbb R^2$ can be seen simply as a plane (or not if that is the case)? Yes, $\mathbb R^2$ is a vector space (not only with that characteristic) with multiple ...
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40 views

Proof that every subspace is a vector space

I was unable to find a simple proof that a subspace is a vector space. I know that a subspace $S$ is a subset of a vector space, such that: $$\vec 0 \in S\\\vec a + \vec b \in S\\\alpha\vec a \in S$$ ...
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1answer
22 views

Use the orthonormality of $u,v,w$ to write the following vectors as linear combinations of $u,v$ and $w$

Let $V$ be the vector space $\mathbb R^3$ with inner product $$(v,w)=3(v_1w_1)-2(v_1w_2)-2(v_2w_1)+5(v_2w_2)-3(v_2w_3)-3(v_3w_2)+3(v_3w_3)$$ where $v=(v_1,v_2,v_3)$ and $w=(w_1,w_2,w_3)$. Part 1 ...
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33 views

Calculate the tensor product of two vectors

Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis ...
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1answer
33 views

Which of the following are true?

I need to find which of the following are true? $\exists A\in M_{2\times 5}(\mathbb{R})\ni\dim$ of null space of $A$ is $2$ My ans: False as $\dim Null(A)+\dim Im(A)=5\Rightarrow\dim ...
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2answers
43 views

Let $A_{j,k} = \langle x_j, x_k\rangle$. Show $A$ is invertible if and only if $x_1, \ldots, x_n$ are linearly independent.

Let $V$ be a vector space over $\mathbb C$ with inner product $\langle, \rangle$ and let $x_1, \ldots, x_n$ be vectors in $V$. Consider the $n \times n$-matrix $A$ with entries $A_{j,k} = \langle ...
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1answer
47 views

$T$-invariant subspace and minimal polynomial

This is the problem that I am stuck on. Problem: Let $V$ be a finite dimensional vector space and $T: V\rightarrow V$ be a linear transformation. Suppose ...
0
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1answer
36 views

Variant of Picard-Lindelof theorem

Question Let $I=[0,a]$ and define the norm $||f||_{\lambda}=\sup_I |e^{-\lambda x}f(x)|$ for $f\in C(I)$. Let $\phi:\;\mathbb{R}^2\to\mathbb{R}$ satify $|\phi(x,u)-\phi(y,v)|\leq\rho |u-v|$ for all ...
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1answer
24 views

A simple question related to One-to-One function and linear operator

I was stuck in one line derivation about the linear operator-related question: Suppose $T$ is linear operator maps from $\mathbb{R}^n$ to $\mathbb{R}^n$. and let $c>0$ be constant. If for all ...
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1answer
15 views

Possible values of nullity in 4x2 matrix

Let $A$ be a 4 by 2 matrix. Explain why the rows of $A$ must be linearly dependent. What are the possible values of nullity(A)? I understand the first part. I do not understand the second part. The ...
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0answers
16 views

What is the analog of the scalar triple product in two dimensions?

Is there a standard name and/or a notation for the analog of the scalar triple product in two dimensions? Namely, i am interested in the following operation: given two elements $\vec u$ and $\vec v$ ...
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2answers
18 views

arc length, problems to find the limits for t

How do I find the limits for t? (a) Let $C$ be the parametric curve $$r(t) = \frac{t^3}{3}\hat i + t^2\hat j + 2t \hat k$$ Determine the arc length of $C$ between the points $(0, 0, 0)$ and $( 1/3, ...
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3answers
102 views

Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
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2answers
38 views
0
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2answers
18 views

Polynomial Ring of Linear Algebraic Group

During lectures, we defined the Linear Algebraic group as the algebraic set $ GL(V):=k^{n^2}-V(Det) $ Where $V(Det)$ are the matrices with $0$ determinant. Then we proceed by identifying the ...
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1answer
20 views

Find the basis of set given by matrices

In linear space of matrix $2\times 3$ over $C$ we have subspace generated by: $ A= \{{\left[\begin{array}{ccc}i&i&i\\i&0&1\end{array}\right]}$ ...
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3answers
33 views

Canonical isomorphism between $V$ vector space and its second dual $V^{\circ \circ}$

I came a across this when I was reading some book. It says let $V$ a finite dimensional vector space of some field and there is a canonical isomorphism $\phi$ between $V$ and $V^{\circ \circ}$ but ...
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2answers
27 views

Write the Jordan form of an operator

These are the properties that apply to the operator $A$. $k_A(x)=x^4(x-2)^4, d(A)=2, d(A^2)=4, d((A-2I))=2, (d((A-2I)^2)=3$ $d$ denotes the defect. $k_A$ is the characteristic polynomial. I ...
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0answers
59 views

With the help of Schwarz inequality for Vectors. Show that function p1 is linearly dependent on function p2.

the given two functions are : $$p_1:=x^2+x$$ and $$p_2:=x-1$$ the innerproduct is also defined as $$L_2^w[a,b]$$ since the degree of the first polynomila is 2 and the degree of second is 1. And ...
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1answer
19 views

Finding base of a subspace

Find base of a subspace and expand it to the base of $\mathbb{R}^4$ subspace is given by the following system of eqiuations: $ \begin{cases} x_1+2x_2+2x_3+4x_4=0 \\ 2x_1+2x_2+x_3+3x_4=0 \end{cases}$ ...
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1answer
28 views

Disprove that this subset of P3 is not a subspace by using a counterexample

The set of all polynomials with degree 3 plus the zero polynomial. A hint would be appreciated to get me going :)
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0answers
12 views

Show subspace can be rewritten as $n-k$ equations

Prove that every $k$ dimensional subspace $V \subset K^n$ can be described using $n-k$ linear equation. I think about applying Kronecker-Capelli theorem.
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1answer
35 views

Finding the Jordan basis of a linear map

A linear map $A$ is given in the canonical basis with the matrix $$ \begin{bmatrix} -2&0&-2&-2\\ 1&0&1&1\\ -1&1&-1&-1\\ 3&-1&3&3\\ \end{bmatrix} $$ ...
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1answer
29 views

show there exist non zero vector which is linear combination of other

sLet $a_1, \ldots , a_n$ be a basis of linear space $V$ let $W \le V$ be a $k$ dimensional subspace $k \ge 1$ Show for each subset $\displaystyle a_{i_i}, \ldots a_{i_m}$ for $m>n-k$ exist non ...
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0answers
34 views

Check if set of functions is a basis of space

Let $f_a \in R^R$ be function given by $f_a(x)=1$ if $x=a$ and $f_a(x)=0 $ if $x \neq a$ for $a \in R$ Decide if set of functions $f_a$ is a basis of space of functions $R^R$ ? I think I know how to ...
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2answers
40 views

Help understanding a proof about vector spaces

The exercise goes like this: -Let $W= {(x,y,z)|2x+3y-z=0}$ Then $W\subseteq\mathbb{R}^3$, find the dimension of $W$. -Find the dimension $[\mathbb{R}^3|W]$ This was a problem from my algebra exam, ...
4
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1answer
93 views

Which rings containing the complex field are, as vector spaces over that field, isomorphic to $\mathbb{C}^2$?

Which rings $R$ containing (as a subring) the complex field $\mathbb{C}$ are, as a vector space over that field, isomorphic to $\mathbb{C}^2$? In other words: what are the two-dimensional unital ...
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0answers
60 views

Normalizing Vectors to get short numbers

$\vec{A}$ is vector agent, $\vec{O}$ is vector Object, $m$ is a constant integer. The following expression is repeated with a different O for every loop cycle: ...
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0answers
18 views

Proof Explanation: Vector Space of Polynomials with Average Value 0 around a circle

The question is from Putnam 2009 B4. Problem: Say that a polynomial with real coefficients in two variable, $x,y$, is balanced if the average value of the polynomial on each circle centered at the ...
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2answers
83 views

Prove $W \cap W^\perp =\{\vec{0}\}$

If $W$ is a subspace of $\mathbb{R}^n$, then $W^\perp = \overline{W} = \{v \cdot w = 0, \forall w \in W\}$ Prove $W \cap W^\perp = \{\vec{0}\}$. How do I fully prove this intersection is ...
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3answers
27 views

Determining if a set is in the subspace of a continuous function

Let $A={\rm span}\{\cos^2x,\sin^2x\}$ be a subspace of the set of functions $C[0,\pi]$, for each of the following functions in $C[0,\pi]$, determine whether or not it is in $A$. $f(x)=1$ ...
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0answers
34 views

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$. I've already proved that if $U$ is a closed subspace then $U = (U^{\bot})^{\bot}$. I also ...
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1answer
23 views

Need help regarding Subspace of matrix and its basis

I need some kind of hint to get me going for this question as I'm so lost at it. Any sort of help would be appreciated. Let E be the set of all 2x2 matrices that have $v={(1,-1)}$ as an eigenvector. ...
1
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1answer
26 views

Find all unit vectors in the plane determined by vectors u and v that are perpendicular to the vector w.

Find all unit vectors in the plane determined by vectors u=(0,1,1) and v=(2,-1,3) that are perpendicular to the vector w=(5,7,-4). This is the question. I found the plane that determined by u and v, ...
0
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1answer
33 views

Basis for solution space?

For the matrix: $$ \begin{bmatrix} 1 & 0 & 2 & | & 0 \\ 0 & 1 & 3 & | & 0 \\ 0 & 0 & 0 & | & 0 \end{bmatrix} $$ ...
2
votes
1answer
20 views

Do two isomorphic finite field extensions have the same dimension?

If $E = F(u_1, \cdots u_n) \cong \bar{E} = F(v_1, \cdots v_m)$ then do the two extensions necessarily have the same dimension over $F$?
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2answers
72 views

Is the statement “the empty set is a subspace of every vector space” true of false?

Is this statement true or false, and why? The empty set is a subspace of every vector space.
2
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2answers
373 views

Exploring underdetermined linear system with non-negative solution

I haven't had much luck searching for this specific problems. Any pointers would be greatly appreciated. I have an underdetermined system where $ A $ and $ b $ are known. $ x $ is a real vector with ...
0
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1answer
38 views

Coordinates of vectors in bases

Two vectors from the standard basis are a = (1,0,1) and b = (1,1,1). What are the coordinates of these vectors in the basis {(1,2,3),(2,3,1),(3,0,1)}. I am not even sure how to answer this question. ...
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0answers
30 views

Can we show it without involving that $V=V^{**}$ are canonically isomorph?

My text proves the following Theorem. Let $V$ be a vector space over $F$ and $B=\{ v_1, \ldots , v_n \}$ a basis of $V$. Then there is exactly one basis $B^*=\{ f_1, \ldots , f_n \}$ of $V^*$ with ...
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0answers
18 views

1st isomorphism theorem for linear transformations (algebra)

For a field K, U' and U'' are vector subspaces of a vector space U over K. It needs to be proven that the transformation φ: U' →(U' +U'')/U'', u' 􏰀→u' +U'', is a surjective linear transformation, ...
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1answer
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Verifying the axioms of a vector space for $V =\{(a,b):a,b\in\mathbb R \}$ with unusual scalar multiplication [closed]

Let $V =\{(a,b):a,b\in\mathbb R \}$. Addition in $V$ is $(a_1,b_1) +(a_2,b_2) = (a_1+a_2, b_1+b_2)$ and scalar multiplication is $k(a,b) = (ka, 0)$. Is $V$ a vector space? Why? I'm mostly lost ...
2
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1answer
37 views

Which Field Operators Construct the Vector Space

Question 14 in F-I-S section 1.2 asks: Let $\mathbf{V}=\{(a_1,a_2,\ldots ,a_n)\colon a_i\in \mathbb{C}$ for $i=1,2,\ldots n\}$; so $\mathbf{V}$ is a vector space over $\mathbb{C}$. Is $\mathbf{V}$ ...
0
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1answer
16 views

Finding vector when conditions are given

Given subspace (of $\mathbb{R}^4$) $V= \rm span ([2,3,1,2], [3,2,2,3], [1,-1,1,1]) $ For $\beta_1=[1,1,1,1], \beta _2=[2,-1,1,2]$ desribe set of all vectors $[b_1, b_2] \in \mathbb{R}^2 $ such that ...
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1answer
18 views

“Absolutely equal” linear functionals and collinearity

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb C$ and let $X^*$ denote its dual (i.e., the space of all continuous linear complex-valued functions over $X$). Suppose that $f,g\in X^*$ ...
1
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1answer
213 views

Prove that the subset $X$ of a normed vector space $(V,\|\cdot\|)$ is complete.

My subset $X$ has the Bolzano-Weierstrass property and I need to prove that $X$ is complete in the sense that every Cauchy sequence in $X$ converges to a point in $X$. I know that having the ...
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0answers
32 views

Independence in Banach space

Everyone knows one of the basic theorems in linear algebra: $k+1$ vectors can't be linear independent in the span of $k$ vectors. Also, it's pretty easy to prove that there is no uncountable system of ...
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0answers
10 views

Axes of rotation, recursive tree branching and GLrotate (computer graphics)

The question is to solve a computer graphics problem, but is essentially a vector math problem so I think it belongs here. My problem is this: a recursive tree is being generated for n iterations ...
0
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0answers
11 views

Why is the following progression allowed?

I'm looking at a proof of a thing related to vector spaces and in that proof we have the following progression: x$^{H}$($\alpha$y + $\beta$z) = $\alpha$x$^{H}$y + $\beta$x$^{H}$z where x, y, z are ...
0
votes
1answer
268 views

Orthogonal Complements and Subspaces Proof

I'm having a little difficulty understanding the proof for orthogonal complements. I kind of understand orthogonal complements, but I cannot seem to find a logic to this. I'm trying to follow along ...