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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Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$.

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$. I know there are lots of answers you could write, but would this be correct: $\{(x-2)^5, (x-2)^4, ...
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0answers
17 views

Vector dimensionality inference

If $\vec{u}$, $\vec{v}$ and $\vec{w}$ are vectors such that: $\vec{u}$ . $\vec{v}$ $\neq 0$ $\vec{u}$ . $\vec{w}$ $\neq 0$ $\vec{v}$ . $\vec{w}$ $\neq 0$ What can be said about the dimensionality ...
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1answer
24 views

If $\dim V = n$ and $S$ is a linearly independent set in $V$, then $S$ is a basis for $V$, True or False?

I'm currently taking a linear algebra course, and the topic of the current section is dimensions of vector spaces. I came across the titled question in the practice problems. Online sources that I've ...
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1answer
7 views

Vector Subspace involving polynomials

H={p(x)∈P2|p(1)=0} is a vector subspace of P2. What is a basis for for H and the dim(H)? I think the dimension is 0 since th restriction of p(1)=0, is that wrong because it is a polynomial?
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Algebraic extensions help?

$K$ is an extension field of $F$. If $[K : F]$ is finite and $u$ is algebraic over $K$, prove that $[F(u) : F]$ divides $[K(u) : F]$.
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1answer
27 views

Is the point on the left or the right of the vector in 2D space?

I'm trying to find if one point on the left or the right of a 2D vector. Example, looking to the figure below; I have the 2D points for a,b and c in the two cases. I'm try to find whether c is located ...
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2answers
17 views

How to find the orthogonal complement of a subspace?

I am having a hard time understanding how to find the orthogonal complement of a subspace $M$ of a vector space $V$. From my modest understanding, $M^\perp$ is a subspace of $V$ where all its ...
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2answers
42 views

Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: ...
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0answers
19 views

Prove that there exits an automorphism from $G$ to $G$ when dim G=infinite

Suppose $G$ is a vector space over $\mathbb Z_2$ . The problem is to prove that there exits an automorphism from $G$ to $G$ Now $G$ has a basis say $\{b_1,b_2,...,b_n\}$.Then any $g\in G$ can be ...
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1answer
44 views

Determining diagonalizability of a matrix containing complex enteries

$$A=\left[\begin{matrix}3-8i&-11+7i\\-1-4i&-2+6i\end{matrix}\right]$$ I've determined the $tr(A) = 1-2i$, and the $det(A)=3-3i$. From here I should be able to use the characteristic equation ...
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1answer
34 views

Finding linear dependance of a set of functions

where the set $B = \{1+2x+2x^2-x^3,3+2x+x^2+x^3,2x^2+2x^3\}$, how can I show they are linearly independent? Could I set the three vectors, u, v, w, into a coefficient matrix and find it's ...
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2answers
21 views

How to show a set of vectors are a basic for a given plane

To determine if a set, B, of the vectors, u, and v for a basis for the plane, W. let u=(1,2,-1), and v=(1,1,1), W =-3x+2y+z=0 I was able to determine the two vectors, w[1], and w[2], from s and t ...
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0answers
19 views

Find the possible signatures of the bilinear forms

Find the possible signatures of the following bilinear forms: The bilinear form $\phi:\mathbb R^n\times\mathbb R^n\to\mathbb R$ given by $\phi(x,y)=x^Tp(A)y$ where $p(t)=t^2+bt+c$ is a ...
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1answer
20 views

If $H$ is a bilinear form then for every $x$ there exists non-null $y$ with $H(x,y)=0$

Prove or disprove: Suppose $H$ is a bilinear form on a finite dimensional vector space $V$, with $\dim(V)>1$. Then for any $x\in V$ there always exists a non-zero $y\in V$ such that $H(x,y)=0$. ...
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0answers
20 views

How to solve matrix differential equations problem

I need to solve $$\begin{bmatrix} 20 & 6 \\ 6 & 7 \end{bmatrix} \begin{bmatrix} y''_1 \\ y''_2 \end{bmatrix} = \begin{bmatrix} 40 \\ 15 \end{bmatrix}$$ And I'm ...
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1answer
56 views

Prove that the isomorphism between vector spaces and their duals is not natural [duplicate]

In preparation for an introductory talk on category theory, I recently spent some time thinking about natural transformations. The first example, or maybe the second, that everyone gives to motivate ...
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0answers
10 views

graph has no bridge iff a spanning subgraph of the graph is the support of a flow

A $\textit{bridge}$ of a graph $G=(V,E)$ (finite graph and we allow loops and multiple edges) is an edge $e$ whose removal disconnects $G$. Let $\mathcal{O}$ be an orientation of the edges of $G$. ...
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1answer
24 views

How do I change this basis for a transformation?

I have $$\left[ L\right]_\mathcal{B}^\mathcal{B} = \begin{pmatrix}2&2&-1\\7&4&-2\\8&5&2\end{pmatrix}$$ and I want to get $[L]_\mathcal{E}^\mathcal{E}$ where the ...
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26 views

3D vector perpendicular calculation [on hold]

Three points $A(6,7,-6)$,$ B(0,0,0)$ and $C(2,6,9)$ are given which are the vertices of a cubes. Find the coordinates of another vertex not on the $ABCD$ plane. I found the answer by finding the ...
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1answer
50 views

Computing the dimension of a vector space in terms of matrix rank

Let $V=\mathbb C^n$ be a complex vector space, and $A,B:V\to V$ two commuting endomorphisms. I am interested in determining the dimension of the vector space $$F_{AB}=\{(a,b)\in V\times V\,|\,A\cdot ...
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1answer
40 views

linear algebra unclear terminology on direct sum

I was given this question in a linear algebra assignment It tells us that V is a vector space over an infinite field F and $ W \subset V $ is a non trivial subspace of V (neither $ V $ nor the the ...
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0answers
23 views

How do I find a vector given by the reflection of a vector with respect to a plane in R^3?

$L: \Bbb{R}^3 \to \Bbb{R}^3$ is reflection with respect to a plane. Two vectors on the plane are given and I calculated their cross product to get the basis $\mathcal{B} = ...
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3answers
359 views

Prove that this vector space is not finite dimensional. [duplicate]

Let $V$ be the set of real numbers. Regard V as a vector space over the field of rational numbers $F$ with the usual operations. Prove that this vector space is not finite dimensional. My attempt: Let ...
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2answers
35 views

I'm having a conceptual issue with similarity matrices

So I know that $A = T^{-1}AT \implies T \text{ is a similarity transformation matrix}$. Say $A = \begin{pmatrix}9 & 13 \\ -3 & -3\end{pmatrix}$, then how would I go about finding T without ...
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0answers
17 views

Showing certain Intersection of Sets equals Closure

given a basis r_1, ..., r_n of an euclidean vector space V with an inner product I would like to show that the closure of S = "the intersection of all sets {x |(x, r_i) > 0} for i=1,...,n" equals ...
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2answers
27 views

Vector spaces with complex field as scalar. [duplicate]

Sorry for stating the question informally. If we have a vector space whose scalars are the field $\mathbb{R}$, if we change the field to be $\mathbb{C}$ and "adapt" the addition and scalar ...
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2answers
52 views

Find the basis for a vector space?

Find the basis of $ V=\{(a, b, c) \in \mathbb {R}^3 \mid a+2b-2c=0\}$ I got $(2, 0, 1)$, $(-2, 1, 0)$ as an answer, but I though that $\dim (V) $ has to be equal to $ n $ for $\mathbb {R}^n $? ...
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1answer
34 views

Properties of dual linear transformation

Let $V,W$ finite dimensional vector spaces over the field $F$, and let $T:V\rightarrow W$ a linear transformation. Define $T^* : W^* \rightarrow V^*$ the dual linear transformation. i.e ...
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1answer
33 views

Question on direct sum of vector spaces

I have the following linear algebra question on direct sums: I am given the vector spaces: $ V = R_4[x] $ $ W = span\{x^4-x^2,3x^4-x^3+1 \}$ I am asked to find the complement to the direct sum i.e. ...
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0answers
19 views

transformation matrices for linear transformation in different basis

I'm working on a problem for my linear algebra class and I can't seem to figure out how this works. The problem states: Let $$L: M_{22} \to M_{22} $$ where $$ L(A) = \begin{pmatrix} 1 & 2 \\ 3 ...
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0answers
49 views

All infinite dimensional vector spaces with a countable basis are space of sequences or functions?

I'm searching a proof that every infinite dimensional vector space that has a countable basis (Schauder basis), can be represented as a space of functions or sequences. All vector spaces of this ...
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0answers
14 views

determine vector equation of the tangent line to a graph at an indicated point

Can you guys help me find a vector equation of the tangent line to a graph at point (6,2,0)? Given that the vector equation of the graph is $$G(x(t),y(t),z(t))=3 \cdot \csc \left({\pi \over 12} ...
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0answers
10 views

To $L_1$ normalize or $L_2$ normalize

I am currently developing a weight function for some AI-related work. Let this weight function $f$ be a linear functional, that is $f(\vec{x}) = \vec{a}\cdot\vec{x}$, where $\vec{a}$ is a coefficient ...
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1answer
19 views

Convex hull of set of sparse vectors?

I am trying to understand how one can define the convex hull of sparse vectors. I understand that for k sparse vectors can be described as a union of subspaces (such as in: ...
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2answers
38 views

Producing for a direct sum $V = U \oplus W$ an endomorphism $P$ of $V$ s.t. $Im(P) = U$ and $\ker(P) = W$

I've been asked to prove this: Let $V$ be a vector space over a field $F$ and suppose that $V = U \oplus W$, where $U$ and $W$ are subspaces of $V$. Show that $Im(P)=U$ and $\ker(P)=W$ for some ...
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1answer
18 views

Dual Vectors and Dual Metric

In the book of Nadir Jeevanjee „An Introduction to Tensors and Group Theory for Physicists“ it is stated as an exercise that: 2.17 Given a basis $\{e_i\}_{i=1,...,n}$ , under what circumstances do ...
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1answer
78 views

What's up with this endofunctor $\mathbf{Aff}_k \rightarrow \mathbf{Aff}_k$?

(Work over a fixed field $k$.) The nLab offers a list of definitions of the concept "affine space". Here's two of them: An affine space is a set $A$ together with a vector space $V$ and an ...
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0answers
16 views

Vector Addition and Subtraction - interpretation

If we have two vectors a and b, both in R^n is it correct to think of 1) a-b as how similar the two vectors are? 2) a + b as moving the vector a in the direction of vector b?
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2answers
39 views

Proving that $\{v_1, \dots v_4\} $ is a basis [closed]

Suppose that $v_1 ,v_2, v_3, v_4$ are non-zero vectors in $R^4$, s.t. $\{v_1,v_2\}$ are independent, $\{v_3,v_4\}$ are independent, and such that both $\{v_1,v_2\}$ are orthogonal to $\{v_3,v_4\}$. ...
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2answers
77 views

Infinite direct product of fields.

Let $F$ be a field, and consider the infinite direct product$$F \times F \times F \times F \times \dots,$$i.e. $\prod_{i=0}^\infty F$, i.e. the direct product of a countable number of copies of $F$. ...
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0answers
14 views

Cokernel of Matrix given only ref($A$)

I am writing down a crib sheet for an exam and I am trying to figure out, if possible, how to determine the cokernel of a matrix, $A$, given only $U$ = ref$(A)$. For example, how might I solve this ...
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1answer
26 views

Diagonalization of quadratic form on a vector subspace

Lets say you have a subspace of a vectorspace which is defined such that $S=\{x: x^Tg_i=0, i=1,\ldots n\}$, basically the orthogonal compement to the space spanned of the g-vectors. Assume also that ...
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2answers
77 views

Find a function $ f : R^2 → R $ such that $ f(λv) = λf(v) $, $\forall \lambda \in R, v ∈ R^2$, yet $f$ is not linear.

Find a function $ f : R^2 → R $ such that $ f(λv) = λf(v) $, $\forall v ∈ R^2$, yet $f$ is not linear. I've been at this for ages, and can't seem to find a single function. I've tried using all ...
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1answer
30 views

Cardinality of the set of linear discontinuous functionals in a normed space

How does one show (or disprove) that for any infinite-dimensional normed vector space $V$, there are uncountably many linearly independent elements in $V^{*}\setminus V'$, where $V^{*}$ and $V'$ ...
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1answer
40 views

Every vector space has a basis

Prove that every vector space has a basis. I am going to use Zorn's lemma for this also here is a necessary definition regarding totally ordered subsets: one element will be contained in the other. ...
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1answer
17 views

Distance from affine vector space?

I've got an affine vector space $W$ defined by a collection of vectors $\{v_1, v_2, ... v_n\}$. Each vector in that space could be represented as a sum of the form $\sum_{i=1}^n w_i * v_i$, where ...
2
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1answer
32 views

Computing bases for direct, wedge, tensor products, etc., of given vector spaces

I am filled with all kinds of vector space and I want to make sure I understand the basis for each kind of vector space. Suppose $\{v_i\}_{i=1}^n$ is the basis for vector space $V$, $\{w_j\}_{j=1}^m$ ...
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1answer
80 views

vector as linear combination of other vectors with one more perpendicular vector

I am reading about Singular Value Decomposition (SVD) from book SVD CSTheory Infoage. At page 6, the chapter says: A matrix $A$ can be described fully by how it transforms the vectors $v_i$. Every ...
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3answers
39 views

Finding the basis of a vector space

Let $V$ be a vector space, and $T : V \to V$ a linear transformation such that $T(2v_1 - 3v_2) = 3v_1 + 5v_2$ and $T(-3v_1 + 5v_2) = -3v_1 + 3v_2$. Then $T(v_1) = ??? v_1 + ??? v_2$ $T(v_2) = ??? ...
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2answers
26 views

Does the trace and determinant uniquely determine the eigenvalues of a 3 by 3 matrix with algebraic multiplicity of 2?

I have a 3 by 3 matrix $M$ whose eigenvalues are $a$, $b$, and $b$. The determinant and trace of $M$ are known from its eigenvalues: $det(M)=ab^2$ and $Tr(M)=2b+a$. I wanted to show that if ...