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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minimum value of a directional derivative

$f=(x^2+y^2+z^2)e^{-(x^2+y^2+z^2)}$ find a point where the direction of the function as a minimum value and is parallel to the vector $3\hat{i}+2\hat{j}+\hat{k}$ So I took $\nabla f=(2xe^{-(x^2+y^2+...
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1answer
23 views

Showing a set is a root system in a vector space from definition of root system

Suppose I have the vectors $\alpha, \beta \in \mathbb{R}^2$ with inner products $(\alpha, \alpha) = 1$ and $(\beta, \beta) = 2$, and the angle between $\alpha$ and $\beta$ is $\theta = \frac{3\pi}{4}$....
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1answer
440 views

Proving any linear transformation can be represented as a matrix

I'm trying to prove that Theorem. Consider a linear transformation $T : \mathbb R^n \to \mathbb R^n$. The transformation $T$ can be represented as a matrix product $\mathbf x \mapsto A \mathbf x$...
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1answer
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What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...
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1answer
27 views

vector generation by linear combination

I have 4 vectors in $R^3$ given as: $v1=(-1,2,0), v2=(3,1,2), v3=(4,-1,0), v4=(0,1,-1)$. I have to show that the vector $v= (5,6,0)$ can be generated by a linear combination of this vector. let the ...
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23 views

Directional derivative

what is the directional derivative of$ f(x,y)=xy+x^2$ at the point $(2,-1,1)$ in the direction $(1,3,-1)$? So the unit vector is $\frac{(1,3,-1)}{\sqrt{11}}$, now we have to take the gradient of ...
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0answers
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Represent of multilinear map [duplicate]

Let $V_1,V_2$ be vector space and $\{e_i\},\{\overline e_i\}$ are basis respectively. $\forall ~l\in L(V_1,V_2; F)$ ,why $l$ can be represented as $$ l=\sum\limits_{ij} a_{ij} \omega^i\otimes \...
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1answer
21 views

Prove that a specific subset $A$ of a nontrivial vector space $V$ over an infinite field $\mathbb{F}$ is infinite

Let $V$ be a nontrivial vector space over an infinite field $\mathbb{F}$. Suppose $V = \bigcup\limits_{i=1}^{n} S_i$, where $S_i$ is a proper subspace of $V$. We assume that $S_1$ is not included in $\...
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1answer
168 views

Orthogonal projection question

Consider the (orthogonal) projection $T: \mathbb{R}^3 \to \mathbb{R}^3$ onto the plane $x - y + z = 0$. (a) Find the standard matrix $[T]_S$ for $T$. (b) Find a new basis $B$ so that $[T]_B$ ...
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1answer
24 views

Symmetric, Antisymmetric, and Alternating Bilinearforms form a vector subspace

I have to show that the space of symmetric, the antisymmetric and the alternating bilinear forms each form a vector subspace of the space of all bilinear forms $\operatorname{Bil}(V,K)$ with $V$ being ...
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1answer
123 views

How to find the appropriate weights to maximize the third coordinate while the first two are zeros

Let's assume, that $v_1, ..., v_n \in \mathbb{R}^3 $ and $ \lambda_1, ..., \lambda_n \in [0, 1] $ The $ v_1, ..., v_n $ vectors are given. I have to find the appropriate weights ($ \lambda_1, ..., \...
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3answers
328 views

Polynomial paths

I am studying Vector Spaces and came across the following problem in Artin: If $x(t)$ and $y(t)$ are quadratic polynomials with real coefficients, show that the image of the path $(x(t), y(t))$ is ...
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1answer
27 views

Every Endomorphism is a Sum of two normal Endomorphisms

How do i show this? I knwo the basic properties of normal Endomorphisms like $$\langle L(v),L(w)\rangle = \langle L^*(v),L^*(w)\rangle $$ $$L^*\circ L = L\circ L^*$$ but i don't really know how to ...
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1answer
130 views

Connecting a vector space to its dual - why?

Can someone explain to me - intuitively - why embedding a vector space into its dual should naturally fix its geometry? I mean, I can run the usual statements through my mind - "The injection into the ...
3
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2answers
41 views

dimension of intersection of a family of subspaces [closed]

Let $V$ be a vector space and $\dim V=n$. Let $U_1,\dots,U_t$ be a family of subspaces of $V$. Assume that if you choose any $n$ subspaces from $\{U_1,\dots,U_t\}$, the dimension of their ...
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234 views

Proof: Sum of dimension of orthogonal complement and vector subspace

Let $V$ be a finite dimensional real vector space with inner product $\langle \, , \rangle$ and let $W$ be a subspace of $V$. The orthogonal complement of $W$ is defined as $$ W^\perp= \left\{ v \in ...
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2answers
36 views

subspace of positive entries [closed]

Can someone show me how to solve this question. Is the set of all vectors in $R^3$ with strictly positive entry a subspace? Thanks
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1answer
33 views

Angle between planes challenging Question

The plane $r.(a,3,5)=10$ is inclined at an angle of $45^\circ$ to the plane $r.(-5,1,4)$ Find the value(s) of $a$ up to $2$ decimal places. I attempted this problem by forming an equation where I ...
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0answers
11 views

How to find contravarient components in this example?

I am wondering how to ffnd contravarient basis vector in terms of covariant in this example on this Wikipedia page https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors#...
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3answers
1k views

$k[x]$-module and cyclic module over a finite dimensional vector space

Given a finite dimensional vector space $V$ over a field $k$ and a linear transformation $T: V \rightarrow V$ we can make $V$ a $k[x]$-module via the map: $$(a_{0}+a_{1}x+\cdots+a_{n}x^{n}) \cdot v \...
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0answers
17 views

conditions of formation of a vector space

Is it compulsory for a linear system to be homogeneous so that the solution space of that system can form a vector space? for example:will the solution space of this linear system form a vector space
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25 views

Is complex multiplication the only multiplication operation on $\mathbb{R}^2$ that works with the Euclidean norm?

What I'm asking is: viewing complex multiplication as binary operation on $\mathbb{R}^2$, is usual multiplication of complex numbers the only operation $\otimes$ on two vectors $\vec{u}$ and $\vec{v} \...
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1answer
16 views

Defining covectors when the basis is oblique

Given a $2$-dimensional vector space with an oblique unit length basis, say, $(f_1, f_2)$, what is the dual vector or covector corresponding to $f_1$, call it $\hat f_1$? There appear to me to be ...
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1answer
34 views

Is the space $\mathbb{R}_+\times S\times S$ linear?

The space $\mathbb{C}$ (or even $\mathbb{R}^2$), which is a linear space over $\mathbb{R}$, can be obtained from the Cartesian product $\mathbb{R}_+\times S$ by gluing to the point the layer $0 \...
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1answer
15 views

Finding a plane which contains a certain line and is perpendicular to another

I have a question on my worksheet which reads the following: 2 lines are given. g1: x= (3,1,3) + t(1,2,-2) and g2: x=(-2,1,-1) + s(0,1,1) the first part is asking if the 2 lines intersect and if they ...
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2answers
27 views

A special linear transformation

Does there exist infinite dimensional vector space V with a linear transformation on V such that nullityT = rank T = dimV
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2answers
56 views

Properties of a $3 × 3$ matrix $A$ that contains two equal rows.

A $3 × 3$ matrix $A$ contains two equal rows. State whether each of the following is true or false. (a) $A$ has an inverse. (b) The rows of $A$ are linearly independent vectors. (c) The determinant ...
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1answer
22 views

Finding orthonormal basis of a polynomial (without a given dimension)

I have the following problem: Find an orthonormal basis of the plane $x_1 + 6x_2 - x_3 = 0$. I am suspecting that this problem is asking me to apply Gram-Schmidt so I can get $q'_1$ and $q'_2$. ...
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2answers
28 views

Find linearly independent vectors formally

How can I find $3$ vectors $a$, $b$ and $c$ in $\mathbb R^3$ such that $\{a, b\}$, $\{a, c\}$ and $\{b, c\}$ are each linearly independent sets of vectors, but the set $\{a, b, c\}$ is linearly ...
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1answer
27 views

Intersection of linear Transformation.

Good night, i was thinking about this: If $T(L_{1}+L_{2})=T(L_{1})+T(L_{2})$, then i can work with this: $T(L_{1}\cap L_{2})=T(L_{1})\cap T(L_{2})$ but I can not think of anything to prove this ...
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1answer
16 views

Which one is the correct definition of natural norm?

In the definition 2 of Normal Subgroup Reconstruction and Quantum Computation Using Group Representations, the authors have defined the natural norm of a matrix as follows. The natural norm of the ...
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1answer
34 views

Cut Space of Vertices without Orthogonal Complement of Cycle Space?

I am studying sparse graphs where their complements tend to be dense (not sparse). I understand this so that the sparse graph has a sparse adjacency matrix while its graph complement is not most ...
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2answers
84 views

Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$

Let $V=\mathbb{C^2}$ be the standard representation of $SL_2(\mathbb{R})$ Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$ I will just consider $SL_2(\...
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2answers
58 views

show that a vector is on a line

can you please explain this question to me? Show that for any two vectors (vector a) and (vector b), the (vector a + vector b)/2 is on the line that connects the vector a and the vector b. show that ...
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3answers
38 views

Distance between a plane and a point

I understood that for finding a distance between a plane and a point we first find a vector between a point on a plane and the given point and then take the projection on the normal vector. Is $D=\...
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5answers
229 views

If $A^2=A$ then prove that $\textrm{tr}(A)=\textrm{rank}(A)$.

Let $A\not=I_n$ be an $n\times n$ matrix such that $A^2=A$ , where $I_n$ is the identity matrix of order $n$. Then prove that , (A) $\textrm{tr}(A)=\textrm{rank}(A)$. (B) $\textrm{rank}(A)+\textrm{...
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1answer
19 views

Distance between two lines

Find the distance between the lines $l_1:$ $x=1+4t,y=5-4t,z=-1+5t$ and $l_2:x=2+8t,y=4-3t,z=5+t$ So the approach in general is to find a vector that is orthogonal to 2 planes that the lines are in ...
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2answers
37 views

How will the basis vectors of the subspace $\mathbb{R}^n$ consisting of those vectors $A=(a_1,\cdots,a_n)$ such that $a_1+\cdots+a_n=0$ look like?

How will the basis vectors of the subspace $\mathbb{R}^n$ consisting of those vectors $A=(a_1,\cdots,a_n)$ such that $a_1+\cdots+a_n=0$ look like? The initial problem was "what is the dimension of ...
0
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1answer
35 views

Compactness in a vector space

If $E$ is a normed space and $F$ is a subspace of $E$, how to prove that if $F\neq\{0\}$ then $F$ is not compact? I begin by this let $x\in F$ then $F=\bigcup_{x\in F} B(x,\varepsilon)$ how to say ...
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1answer
57 views

Determine if the Set of Vector is a Subspace of $\mathbb{R}^n$

Can you help me check whether what I did is right or wrong? Here are the questions: Which of the following sets are subspaces of $\mathbb{R}^n$? (a) The plane of $\mathbb{R}^3$ that passes ...
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3answers
41 views

What is the space $\operatorname{Sym}^2(V)$ and how does it act on the vector space $V$?

If $V$ is a vector space over $\mathbb{C}$ with basis vectors $e_i$, what is the space $\operatorname{Sym}^2(V)$? I am hoping someone can give me some insight into this space; perhaps by describing ...
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0answers
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Representation decomposition over $GL_2(\mathbb{C})$

I have found that $Sym^2(V) \otimes Sym^2(V)$ can be decomposed over the special linear group as follows: $Sym^2(V) \otimes Sym^2(V) \simeq Sym^4(V) \oplus Sym^2(V) \oplus 1$ This is done using the ...
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1answer
20 views

Understanding the dimension of a particular subspace in Linear Algebra in C³

I am having trouble to determine the dimension of the subspace of T. How is it done, when there is just one vector given?
2
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1answer
69 views

Getting an isomorphism from a short exact sequence of inner product spaces

Let $L,M,N$ be finite dimensional inner product spaces and $0 \to L \xrightarrow{\alpha} M \xrightarrow{\beta} N \to 0$ is a short exact sequence. Now let $\beta^* : N \to M $ be the adjoint map (the ...
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0answers
13 views

Free graded k[x] modules have homogeneous bases

I was reading the article "Cary Webb. Decomposition of graded modules. Proceedings of the American Math- ematical Society, 94(4):565–571, 1985" where in the beginning "Free graded k[x] modules have ...
2
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1answer
32 views

Can you construct a basis for an infinite dimensional vector space from a set of vectors that span that space?

Suppose I have an infinite dimension vector space V (not necessarily countably infinite). Suppose a have a set S that spans the space. If V is finite dimensional, it is trivial to construct a basis ...
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1answer
79 views

In an infinite dimensional real inner-product space , can any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis?

Let $V$ be an infinite dimensional real inner-product space , then is it true that any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis ? Or at least is it true ...
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2answers
41 views

geometic description of vector span

i am having some poblem with this question Show that the set if vector is linearly dependant. If the vector span a line, provide the equation of the line. If the vector span a plane, provide the ...
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0answers
25 views

Suppose we have three vectors $x,y,z\in \mathbb{R^3}$

Suppose we have three vectors $x,y,z\in \mathbb{R^3}$ Further suppose that x ⊥ y and y ⊥ z then is x ⊥ z?
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1answer
46 views

Prove that this transformation inverse exists and it's bounded

If $X,Y$ are Normed Vectorial Spaces, $T$ is a bounded lineal transformation. Prove that if exists $b>0$ such that $\|Tx\|\geq b\|x\| \forall x\in X$. Then $T^{-1}:Y\rightarrow X$ exists and it's ...