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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The set of all $n\times n$ matrices A such that the $A^T = A^{-1}$ is a subspace of the vertor space $M_n(\mathbb{R})$

I think the set of $n \times n$ matrices such that $A^T = A^{-1}$ is not a vector space since it doesn't have $0$. How do I show that it's not a subspace?
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Is the converse of the Pythagorean Theorem false for complex inner products?

I was thinking about the converse of the Pythagorean theorem: $\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y\rVert^2 \implies x \perp y$ Does this hold if the inner product $\langle \cdot,\...
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Simple excercise on linear transformations - confused

A Linear tranformation L in $\mathbb R^3$ with matrix $$ L_b^b = \left(\begin{matrix} 1 & 0 & 5 \\ 0 & -2 & 2 \\ 1 & -2 & 7 \end{matrix}\right)$$ and basis $b = \{ (1,0,2), (0,...
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Dimension of subspace of $\text{End}(\mathbb{R}^5)$

I'm doing a problem which presented me with a basis for some $U\subseteq\mathbb{R}^5$ where $\dim U=3$ (I can give it explicitly if that helps but I do not think it matters). The question is this: ...
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Gröbner basis is not a vector basis?

We use the same notation for Gröbner basis and vector basis. I recall that $\langle 1\rangle_{GR}$ is the largest Gröbner basis while $\langle 1\rangle_{vector}$ is the smallest vector basis. So for ...
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Basis of all real polynomials?

I am studying the book Topics in Algebraic Graph Theory by Beineke et all and the page 12. By the book, the set of all real polynomials can be generated by the set $\{1,x,x^2,\ldots\}$ which I ...
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Matrix equivalent to linear maps - sanity check

I'm reading some Linear algebra notes I found online, and am a bit confused about the following: If $U,V$ are finite dimensional $\mathbb{C}$-spaces with bases $(\mathbf{u}_1,\dots,\mathbf{u}_m)$ and ...
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Terminology: If $A, B$ are subspaces of $V$ and $A \cap B = \{0\}$ then they are …?

If $A, B$ are subspaces of $V$ and $A \cap B = \{0\}$ then ... If $V = A \oplus B$ they are complementary, otherwise I think that Halmos describes them as disjoint but this seems at odds with the ...
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Understanding a proof of a theorem from S.Roman's “Advanced Linear Algebra”

There is a Theorem $1.5$ on page $43$ of the book "Advanced Linear Algebra" by Steven Roman. Theorem $1.5$. Let $F = \{ S_i | i \in I \}$ be a family of distinc subspaces of a vector space $V$. ...
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Infinite matrix product

Let $$X=\left(\begin{array}{c} x_1 \\ x_2\\ \vdots \end{array}\right)$$ be an infinite real vector and $$A=(a_{ij}), \ 0<i,j<\infty$$ be an infinite real matrix. (1) For which $A$ can one ...
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For a Vector Space $V = A \oplus B = A \oplus C \implies dim(B) = dim(C) $?

For a finite dimensional space there is no problem. $dim(V) = dim(A) + dim(B) = dim(A) + dim(C) \implies dim(B) = dim(C)$ For an infinite dimensional space it still holds that $dim(V) = dim(A) + ...
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angle between two planes, why can we use the dot product?

I understand that to find the angle we use the dot product of the normal vectors of the two planes, but why is it correct? as the normal vectors are both 90 degrees from the "real" angle of the planes
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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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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

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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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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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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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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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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59 views

derivative of a vector

find $\frac{d^2\vec{S}}{dt}$ where $\vec{S}=(t+1)\hat{i}+(t^2+t+1)\hat{j}+(t^3+t^2+t)\hat{k}$ So $\frac{d\vec{S}}{dt}=\hat{i}+(2t+1)\hat{j}+(3t^2+2t+1)\hat{k}$ now when I take the derivative again ...
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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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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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63 views

How to compute the projection of a vector on a plane

Can someone check whether my work is correct or not? Compute the projection of $(1,1,1)$ onto the plane that passes through the points $(1,0,-1), (3,7,-3), (-2,-1,2)$. My attempt: Let $u = (1,1,...
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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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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
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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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17 views

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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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?
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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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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 ...
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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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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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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 ...