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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Charactristic polynomial of a F-linear transformation with respect to Galois group

Let $K$ be a Galois extension of $F$, and let $a \in K$. Let $L_a : K \to K$ be the $F$-linear transformation defined by $L_a(b)=ab$. Show that the characteristic polynomial of $L_a$ is $\prod_{\sigma ...
2
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1answer
51 views

A question about Matrices and Linear Transfromations

Let $v_1,...,v_n$ be a basis of a vector space $V$ over a field $K$. Let $M(T)$ denote the matrix of a linear map $T:V \rightarrow V$ with respect to our basis. Prove $$M(ST)=M(S)M(T)$$ for all ...
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1answer
145 views

If $g:V \rightarrow V$ is an injective linear transformation. Prove if $V$ is finite dimensional then $g$ is surjective.

I am asked to prove this without the rank nullity theorem My Attempt at a Proof For the $\implies$; If $g:V \rightarrow V$ is injective then the dimension of the kernel is 0, and so as ($im$) ...
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459 views

What is the operation inverse to vectorization (vec operator)?

There is a well knows vectorization operation in matrix analysis $\mbox{vec}$: https://en.wikipedia.org/wiki/Vectorization_%28mathematics%29 I've vectorized my matrix equations, did some ...
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1answer
41 views

determine the kernel of this function, find a basis for the kernel and the dual space?

So i was having trouble with this and would really, really appreciate if people could help me out because i'm getting stressed out about it. here's the question: so let $P_3$ denote the vector ...
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3answers
37 views

find a base to U Linear Algebra

dear users please help me... im answering a long question now ive been guided to find a base to U at the end of the process i got this $u= Sp\{x^4-3x^3+2x^2, 3x^4-7x^3+4x ,1\}$ and ive been guided to ...
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2answers
80 views

What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
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0answers
23 views

Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle. Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy ...
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3answers
52 views

Matrices and Inverses

Need a bit of help with this question. We're given two invertible square $n\times n$ matrices $A$ and $B$ with entries in the reals. We have to show that $AB$ is also invertible and then express ...
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1answer
208 views

matrix inverse in tensor notation

Suppose there is a matrix $A$ that transforms vectors, $$ Y = A x $$ Now express this in some other coordinate system, with $x = B z, \,\, y = B w$, so \begin{align*} & Bw = A B z \\ ...
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18 views

Polynom Space, check if U a base

$R_5[x]$ is a polynom space which is lower than 5 over R (Including the zero polynom). Given: $U = \{p(x) \in R_5[x] | p(0) = p(1) = p(2)\}$ Prove that U is a sub-space of $R_5[x]$. Find a base to ...
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1answer
52 views

Prove that $ A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$

I'd like to get some help So I need to prove that when $A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$ Linear Algebra, of course. Thanks
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2answers
193 views

How to prove this Gram determinant

Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$ \det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ...
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0answers
76 views

Best way to solve specific block-tridiagonal linear system (10000x10000 and larger)

To provide more context, this system came from energy balance equation on a mesh with (n,m) nodes in each direction. It's a linear system that looks like this (size of system in blocks n = 4, size of ...
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36 views

Tensor Product over a field

This question appeared in my exam and I could not solve it. Let $L$ be a field, $K$ subfield of $L$. Assume that dim$_K(L)$=$m$, and $V$ be a $L-$vector space amd dim$_L(V)=n$. If as usual ...
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1answer
44 views

Problem on Matrices (Linear Algebra)

If A is a real orthogonal matrix and $I+A$ is non singular prove that the matrix $(I+A)^{-1}(I-A)$ is skew symetric
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1answer
45 views

Find determinant $\det M$, where $m_{ij}=a_ia_j$, and $m_{ii}=a^2_i+k$

Let ${a_1,\dots,a_n}$ --- sequence and $k\ne 0$. Define matrix $M$ in following way: $m_{ij}=a_ia_j$ if $i\ne j$, and $m_{ii}=a^2_i+k$. Find $\det M$.
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1answer
105 views

Find the two points where the shortest distance occurs on two lines

Find the point P on $\vec{AB}$ and point Q on $\vec{CD}$ such that $\vec{PQ}$ is the shortest distance between the lines AB and CD, given $\vec{AB} = \begin{pmatrix} 1\\ 0\\ 2\\ \end{pmatrix} + ...
2
votes
1answer
88 views

Coordinate-free proof of determinant of transpose

I'm interested in a coordinate-free proof of the statement $\mathrm{det}(A) = \mathrm{det}(A^T).$ Let $V$ be a finite-dimensional vector space over a field $K$, and let $f : V \rightarrow V$ be an ...
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0answers
45 views

Scalable QR decomposition algorithm

Suppose one has a processor for QR decomposition of complex matrix of size 4 x 4. So if it is necessary to decompose M x M complex matrix, A, one can represent it as R x R block matrix [Cij] (block ...
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3answers
75 views

Invertible composition of linear transformation [closed]

If both $L:V\rightarrow W$ and $M:W\rightarrow U$ are linear transformations that are invertible, how can you prove that the composition $(M\circ L):V\rightarrow U$ is also invertible.
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1answer
431 views

How to calculate per unit costs for multiple items

I had a supplier give me a quote last week that seems very strange, can someone help me out? The quote is for IT hardware, but for simplicity (and anonymity) I'll use apples and oranges: ...
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1answer
79 views

An algebra/linear algebra question

Suppose 8 real numbers $a,b,c,d$ and $x,y,z,w$ satisfy \begin{equation*} a^2+b^2+c^2+d^2=x^2+y^2+z^2+w^2=1,\quad ax+by+cz+dw=0. \end{equation*} Is it true that \begin{equation*} ...
2
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2answers
83 views

How to verify whether a solution to an optimization problem is correct.

Consider a general optimization problem min f(x) subject to g(x) <=0 h(x)=0, where x denotes a vector and the functions are $R^n$ -> $R^n$. suppose ...
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1answer
70 views

Simultaneous diagonalizability of three endomorphisms

So I understand two endomorphisms on a finite dimensional vector space are simultaneously diagonalisable if (and only if) they commute. But suppose we have three endomorphisms a, b and c. What is the ...
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votes
3answers
97 views

Matrix inequality for square matrices

Does the following hold for any square matrix $A$, $(AA^*)^{1/2}\geq (A+A^*)/2$, where superscript $*$ denotes the Hermitian transpose. Proof/any comment would be appreciated.
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0answers
38 views

Minimize the number of nonzero elements of a matrix through elementary row operations?

Is there a general method to minimize the number of nonzero elements of a real rectangular matrix through elementary row operations? I am looking for something analogous to Gaussian elimination, that ...
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0answers
117 views

Dimensions of Direct Sum vs Direct Product

I want to show that, if $A$ is an infinite set and $F$ is a field, then the direct sum of copies of $F$ indexed by $A$ has strictly lower dimension than the corresponding direct product. I know that ...
23
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9answers
2k views

Very good linear algebra book.

I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), ...
7
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2answers
127 views

How to find determinant of this matrix?

Is there a manual method to find $\det\left(XY^{-1}\right)$ ? Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\ 1 & 3 & 3^2 & \cdots & 3^{2012} \\ ...
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2answers
51 views

Prove that if $B=P^{-1}AP$, then $q(B)=P^{-1}q(A)P$

Is it possible to prove this using a similarity invariant? For example showing that $$\det(q(B))=\det(q(A))$$
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1answer
32 views

Find subset of rows whose entries sum to an even number in each column

I am trying to implement Fermat factorization with factor bases. The textbook suggests using row-reduction to find a linearly dependent set of rows. How does one go about finding such a linearly ...
0
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1answer
33 views

Find the eigenvalues of…

My characteristic equation starts off: $$\lambda(\lambda(\lambda-3k)+3k^2)-k^3=0$$ Once expanded I get: $$\lambda^3-3\lambda^2k+3\lambda k^2-k^3=0$$ Where do I go from here?
3
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1answer
171 views

Rank one plus diagonal matrix approximation

Given $A \in R^{n \times n}$, $A$ symmetric. I'm trying to solve the following minimization problem: $\underset{u \in R^n, d \in R^n} \min \, \frac{1}{2} \|X - A\|_F^2$ subject to $X = u u^T + ...
0
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1answer
25 views

Help with a linear algebra question (transition matrices)

I can't seem to figure out to do part b). Is the correct procedure to multiply $[V]_s$ by $P$? Can anyone tell me the appropriate procedure?
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2answers
112 views

Deducing formula for nth term in sequence and validate using principles of induction

I a working my way through some old exam papers but have come up with a problem. One question on sequences and induction goes: A sequence of integers $x_1, x_2,\cdots, x_k,\cdots$ is defind ...
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1answer
317 views

how to find matrix A from complete solution to Ax=b

I am trying to solve a problem. I was stuck.Any help is appreciated. The complete solution to $Ax=\left[\begin{array}[c]{rr}1 \\3 \end{array}\right]$ is $ x= ...
3
votes
1answer
100 views

Change of coordinates (referential system) mistake? Doesn't seem to yield the proper coordinates.

Let $\varepsilon$ be an affine space with referential system $R$ characterized by $O=(1,1,1)$ as origin and $B=(c_1,c_2,c_3)$ as its basis, which is the canonical. Now, lets define a new ...
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3answers
41 views

Subspace of $\mathbb{R}^3$

$\{(x,y,z)\mid 2x^2-8y^2-9z^2\le 1\}$ Why is this not a subspace of $\mathbb{R}^3$? I know the zero vector is in this set because $ 2(0) - 8(0) - 9(0) \le 1$ But I can't seem to verify if the ...
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0answers
46 views

Need help in deriving mathematical formula

I need a mathematical formula that would give me the specified result for given input x (x is always an integer) ...
2
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1answer
67 views

How can we prove that the rank of a matrix is a non-convex function of that matrix?

How can we prove that $\operatorname{rank}(\mathbf{X})= 1$ is a non-convex function of $\mathbf{X}$.
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4answers
55 views

Express the inverse of $A^T$ in terms of $A^{-1}$

Let $A ∈ M$ $n×n(R)$ be an invertible matrix. Prove that $A^T$ is also invertible, and express the inverse of $A^T$ in terms of $A^{-1}$. We've already proven the first part, its just the second ...
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1answer
27 views

Writing vectors as linear combinations of bases $e_i\otimes e_j$ and $e_1\wedge e_2,e_1\wedge e_3, e_2\wedge e_3$

Write $$\begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix} \otimes \begin{pmatrix}2 \\ 1 \\ 1\end{pmatrix} + \begin{pmatrix}1 \\ -1 \\ 5\end{pmatrix} \otimes \begin{pmatrix}4 \\ 0 \\ 3\end{pmatrix}$$ as linear ...
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1answer
32 views

the differences and relationship between linear independent and affinely independent

When learning optimization, I heard the two related concepts on linear algebra: linearly independent and affinely independent. ...
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What does the phrase “uncoupled across coordinate directions” mean in this text?

The following paragraph is from a paper about comparison of maneuvering target tracking models.In the paragraph it talks about constant acceleration models. The above models are simple but crude. ...
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1answer
42 views

Linear Algebra. Solution help.

1) Let $P_n$ be the set of polynomials up to the $n$-th degree. Show that $P_n$ is a subspace of the linear space $\def\R{\mathbb{R}}F(\R,\R) = \{ f \mid f : \R \to \R \}$ where $f$ is a function. ...
2
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1answer
149 views

Check if the following gradient is correct

This question regards the verification of the gradient of a given function. Notation. Let $N, K \in \mathbb{N}_0$ be given (nonzero) integers, with $K > N$. Let $\mathbf{x} = [x_b \ y_b \ z_b]^T ...
3
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1answer
31 views

Expressing components of a block matrix

Say you have a block matrix of $n \times n$ matrices $M = \begin{pmatrix} A&B\\ C&D \end{pmatrix}$ and you know rank ($A$) = rank($M$) = $n$. Show that $D= CA^{-1}B$. I'm really confused as to ...
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1answer
121 views

Proving minimum polynomial equals characteristic polynomial in a cyclic vector space

Let $V$ be a vector space over a field $\mathbb{F}$ and $T:V \rightarrow V$ be a linear map. Let $v \in V$ be such that $\left \{ v,T(v),T^{2}(v)... \right \}$ spans $V$. I have proved that $B=\left ...
0
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1answer
22 views

Local boundedness of linear operators

According to this definition of a bounded linear operator L(v): X -> Y the bounding constant M must be the same for all elements of the preimage X of the operator. However it then says that a bounded ...