Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Changing a basis and satisfying inequalities

I have a problem where I must have that $x, y, z \gt 0$, but I have also found that the solution lies on the plane spanned by the vectors: $$\begin{pmatrix} -0.58 \\ 0.79 \\ -0.21 \end{pmatrix}r + ...
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2answers
135 views

How to prove Hom$(U,V)\cong U^∗ \otimes V$

I know this result is well-known, but could some one give me some help with this proof - or some reference? Thank you. Also, I have taken second year linear algebra (Axler's Linear Algebra Done ...
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1answer
202 views

Prove there is a basis-independent isomorphism between $T^1_1(V)$ and the space of linear maps $V \rightarrow V$

Let $V$ be a finite-dimensional vector space. Prove there is a basis-independent isomorphism between $T^1_1(V)$ and the space of linear maps $V \rightarrow V$. I want to show there is a bijective ...
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2answers
30 views

Conversion of price in different seconds to determine the speed of the fall

I'm having a thought problem. I'm looking for a conversion to determine the speed of the price decrease. e.g if \begin{align} \$398 & = 120\text{ seconds} \\ \$62 & = 0\text{ seconds} \\ ?? ...
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1answer
196 views

cauchy schwarz equality: difference in proving style for linear algebra and expectation version

I am interested in proving the following sub version of Cauchy Schawrz equality. 1) LA version : If $x$ and $y$ are two real vectors and the following holds $$<x,y> = ||x||.||y||$$ then $x$ ...
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2answers
86 views

Quick method /Birds eye view to determine the value

Is there any way to guess the answer without doing elaborating calculations?
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1answer
142 views

optimizing a logdet function with respect to a scalar and the Hessian matrix

Given a logdet function $\mathcal{L}(\gamma)$, $$ \mathcal{L}(\gamma) = \log\vert \mathbf{I} + \gamma\mathbf{S} \vert - \mathbf{q}^T(\gamma^{-1}\mathbf{I} + \mathbf{S})^{-1} \mathbf{q}, $$ where ...
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1answer
78 views

How to analyze $A\cdot (\mathop{\rm tri} A)^{-1}$?

Suppose I have an upper triangular square matrix $A$, and $\mathop{\rm tri}A$ is the operator which takes the tridiagonal part of $A$. Assuming that we know ${\rm tri}(A)$ is invertible. I am trying ...
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1answer
145 views

basis-independent isomorphism

Could some one give me some help with this proof? Given the hint, I still don't have a clue about how to proceed. Thanks.
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1answer
177 views

Condition on a matrix sum with equal determinant and trace

Let $n$ be a positive integer, $J$ the matrix of all ones and $Q$ a symmetric positive semidefinite matrix such that $\det(nI-Q) = \det(Q+J)$ $\rm{tr}(nI-Q) = \rm{tr}(Q+J)$ and also $nI-Q \ne ...
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1answer
898 views

Cholesky for Non-Positive Definite Matrices

I am trying to approximate a NPD matrix with the nearest PD form and compute its Cholesky decomposition. I know that the usual method is to perform an eigenvalue decomposition, zero out the negative ...
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1answer
72 views

Lines and planes at space

First of all, sorry for my poor English. Can someone please help me? How can I find the parametric equation of the line that have $ A=(1, -2, -1) $ and passes through the skew lines $r:$ ...
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38 views

charpoly and rank of a projection

$A$ is a projection on $\mathbb{R}^n$ so I know $A^2=A$ and eigen values are $1,0$ given that $\dim Ker(A-I)=m$ then what is the characterist polynomaial of $A$ and rank of $A$? I am not getting ...
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2answers
86 views

Prove $A\in \mathbb R^{n\times n}$ is antisymmetric iff…

Prove that $A\in \mathbb R^{n\times n}$ is antisymmetric iff $ \forall v\in\mathbb R^n:\langle v,Av\rangle=0 $ $\langle \cdot,\cdot\rangle$ is just the dot product. I'm a little stumped by this ...
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1answer
95 views

A Unique Invariant subspace for a set of matrices

Im wondering if anyone can give me a good reference or answer this question which may have already be solved. For a set of generic $n\times n$ matrices $A_1,A_2,...,A_k$, such that they share only ...
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1answer
284 views

Invariant space of linear transformation

Let $V$ be a vector space of a finite nonzero dimension $n$ over some field. Let $T$ be a linear transformation of $V$, such that $T$ is nonzero and not one-to one. (a)Give a $T$-invariant linear ...
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3answers
101 views

Finding eigen values of a binary matrix with diagonal elements are all 0s and non-diagonals are 1s

How to find the eigen values of an $n$x$n$ matrix whose diagonal elements are all 0s and non-diagonal elements are all 1s ? Please don't tell the solution. I just want a hint.
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1answer
75 views

Show the eigenspaces of $T$ are all $1$-dimensional

Let $V$ be a finite-dimensional complex vector space and $T:V\to V$ a linear transformation. Suppose there exists $v\in V$ such that $\{v,Tv,T^2v,\ldots,T^{n-1}v\}$ is a basis for $V$. Show that the ...
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3answers
153 views

Solve matrix equation $XB + CX^{-1} = aI$

I wonder if it's possible to find positive-definite matrix $X$ such that $$XB + CX^{-1} = aI$$ $a$ is known non-negative scalar, matrices $X$, $B$ and $C$ are symmetric and have the same size
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2answers
136 views

How to diagonalize $f(x,y,z)=xy+yz+xz$

Could you tell me how to diagonalize $f(x,y,z)=xy+yz+xz$. I know I can rewrite it as $(x+ \frac{1}{2}y + \frac{1}{2}z)^2 - x^2 - \frac{1}{4}(y-z)^2$ What do I do next? Could you help me?
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1answer
61 views

Quadratic form $\mathbb{R}^n$ homogeneous polynomial degree $2$

Could you help me with the following problem? My definition of a quadratic form is: it is a mapping $h: \ V \rightarrow \mathbb{R}$ such that there exists a bilinear form $\varphi: \ V \times V ...
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1answer
1k views

Proving replacement theorem?

I want to see if I am understanding the proof of the replacement theorem correctly. Let V be a vector space that is generated by a set G containing n vectors. Let L be a linearly independent subset ...
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2answers
222 views

Prove that one of the following sets is a subspace and the other isn't?

OK, here goes another. Prove that $ W_1 = ${$(a_1, a_2, \ldots, a_n) \in F^n : a_1 + a_2 + \cdots + a_n = 0$} is a subspace of $F^n$ but $ W_2 = ${$(a_1, a_2, \ldots, a_n) \in F^n : a_1 + a_2 + ...
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65 views

is this subset a subspace - redux

OK, I have been bothering people here with this for days and with luck I finally have this. People have helped a lot here so far. (Doing these examples is I hope helping me learn the proofs, but I ...
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5answers
158 views

Finding the limit of a matrix

Suppose that $A=\begin{pmatrix}4&1 & 5\\ 2& 7& 1\\ 2& 2& 6\end{pmatrix}.\;$ How can I find $\;\displaystyle \lim_{n\to\infty}A^n$? What theorem(s) should I use to solve this? ...
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2answers
746 views

An operator that commutes with another operator $T$ with distinct characteristic values is a polynomial in $T$

I'm trying to solve some problems in Hoffman and Kunze and I'm kind of stuck on this one. This is 6.5.3 on Hoffman and Kunze. Here is the question: Let $T$ be a a linear operator on an ...
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5answers
1k views

How to prove that the set $\{\sin(x),\sin(2x),…,\sin(mx)\}$ is linearly independent?

Could you help me to show that the functions $\sin(x),\sin(2x),...,\sin(mx)\in V$ are linearly independent, where $V$ is the space of real functions? Thanks.
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3answers
150 views

Suppose that a $3\times 3$ matrix $M$ has an eigenspace of dimension $3$. Prove that $M$ is a diagonal matrix.

How would I go about this? I realise that having dimension 3 means that the solution to $(A-\lambda I)\mathbf b = \mathbf 0$ has 3 free parameters, which would in turn mean that $(A-\lambda I)$ is the ...
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2answers
160 views

if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$? [duplicate]

If $A$ is abelian group and $B$ is a subgroup of $A$, $B$ is normal subgroup of $A$. Is it true that $B \times A/B \cong A$? I ask because I was watching an online lecture from a course in abstract ...
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1answer
209 views

Dimension of the space of algebraic Riemann curvature tensors

Given $n\in \mathbb N$, consider the vector space $\mathbb R^{n^4}$ whose elements I will denote by $(R_{abcd})$ with indices $a,b,c,d \in \{1, \dots, n\}$. This vector space is $n^4$-dimensional. The ...
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1answer
34 views

Relationship between $(L|_M)^*:N^*\to M^*$ and $L^*|_{N^0}:N^0\to M^0$?

Suppose $L:V\to W$ is a linear transformation, and $L(M)\subseteq N$ for some subspaces $M\subseteq V$ and $N\subseteq W$. A question I'm reading asks rather open-endedly if there is a relationship ...
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2answers
806 views

Given a linear transformation matrix, T, find the equation for the curve that T transforms a circle into.

Given the linear transformation matrix: $$T=\pmatrix{2&-3\\1&1}$$ Find the equation for the curve that $T$ transforms a circle with equation $x^2+y^2=6$ into. What I know: My basis is going ...
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2answers
57 views

Derivative of column-row multiplication

How can I take derivative $$\frac{d}{dA}(x - Ab)(x - Ab)^T$$ where $x$ and $b$ are known vectors of the same size and matrix $A$ is symmetric and positive-definite? Update: This expression could be ...
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0answers
281 views

Proof of the spectral theorem for normal operators from two lemmas

I have the following lemmas that I can prove: Let $T$ be a linear operator on a Hermitian space $V$ and let $W$ be a $T$-invariant subspace of $V$ . Then $W^⊥$ is $T^*$-invariant Let $T$ be a normal ...
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155 views

The vector space of polynomials

I was given a theorem: The polynomials (where $f$ and $g$ are complex polynomials of degrees $n$ and $m$) $$f(z), zf(z), \ldots , z^{m−1}f(z), g(z), zg(z), \ldots,z^{n−1}g(z)\tag{7.6.4}$$ ...
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1answer
164 views

How to show all eigenvalues are positive?

Could you help me to show that the following matrix has all its eigenvalues positive? $$H= \begin{bmatrix} \sum_{k=1}^ng_1(x_k)^2 & \sum_{k=1}^ng_1(x_k)g_2(x_k) & \cdots & ...
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1answer
68 views

Proving that a matrix is diagonalizable

Let $ T $ be the linear operator on $ \Bbb R^3 $ which is represented by the matrix $$ A = \begin{bmatrix} 6 & -3 & -2 \\ 4 & -1 & -2 \\ 10 & -5 ...
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1answer
113 views

Determine all functions for a given gradient

I don't really have an approach to solve this problem so it would be very kind if you could tell me what to do first: Determine all functions $f:\mathbb{R}^2 \to \mathbb{R}$ for which applies: ...
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1answer
48 views

eigenvalue and independence

Let $B$ be a $5\times 5$ real matrix and assume: $B$ has eigenvalues 2 and 3 with corresponding eigenvectors $p_1$ and $p_3$, respectively. $B$ has generalized eigenvectors $p_2,p_4$ and $p_5$ ...
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2answers
269 views

Linear algebra T- invariant subspaces

Let $ T $ be the linear operator on $ \Bbb R^3 $ which is represented by the matrix $ A = $ $ \begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & ...
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1answer
74 views

Help with anti-image matrix

First of all, I am very sorry but I don't know the mathematics terminology in English, so I'll try to explain as good as i can but i will probably do some mistakes since it's not my native language. ...
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38 views

Existence of weights of a finite dimensional representation of a semisimple Lie algebra

Let $\mathfrak{g}$ be a semisimple complex Lie algebra. I want to show that every finite dimensional irreducible representation of $\mathfrak{g}$ is a weight module, and I need the existence of at ...
2
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2answers
735 views

How to show that a given set is a subspace

OK I just want to be sure I have done this correctly. Given: $R^3$, are the following sets subspaces? (a) $W_1$ = {($a_1$,$a_2$,$a_3$) $\in R^3: a_1 = 3a_2$ and $a_3 = -a_2$ Since the set you get ...
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2answers
145 views

Show that this set is linearly independent

Ñotation: $V$ is a vector spaces of real functions $g:X\rightarrow\mathbb{R}$; $\{g_1,...,g_m\}$ is a subset of $V$; $\{x_1,...,x_n\}$ is a subset of $X$, where $x_i\neq x_j$ when $i\neq j$; ...
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3answers
65 views

Symmetric Matrices Help

I'm having a bit of trouble with the following question. Suppose $A$ is a square matrix. a) Show that the matrix $B = A+A^T$ is symmetric. Not sure how to do this. But here is my attempt. Well, let ...
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1answer
120 views

Orthonormal basis, isomorphism preserving dot product

For an orthonormal basis $v_1, ..., v_n$ of $(V, \cdot )$ $\mathbb{R}^n \ni (x_1, ..., x_n) \rightarrow \sum x_jv_j \in V$ is an isomorphism preserving dot product. I've already proven that it ...
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3answers
1k views

Are the eigenvalues of $A^\top A$ equal to those of $AA^\top$?

In an exam question I was asked to calculate the eigenvalues of $A^\top A$, where $A = (a_1\ a_2\ a_3); a_1=(0\ 2\ 1)^\top; a_2=(1\ -1\ 1)^\top; a_3=(1\ 1\ -1)^\top;$ and $A^\top$ stands for the ...
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1answer
224 views

Jacobian determinant and orientation

So in Jacobian determinant, it is often said that it gives information about whether Jacobian matrix changes orientation, but I cannot get what orientation exactly in this context.
2
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103 views

Ideals (one-sided ideals) of $n×n$ upper triangular matrices

Is there any characterization of ideals (one-sided ideals) of $n\times n$ upper triangular matrices? I have just seen in monthly journal about $2 \times 2$ matrices in the below article Left and Right ...
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88 views

Finding if the Point is in the triangle

Can You Please tell me if this is right Point(x,y,z) triangle points ABC using co-planer determinant |x-Ax y-Ay z-Az| |x-Bx y-By z-Bz| = 0 |x-Cx y-Cy z-Cz| then the point is in the tringle