Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Proof that $\det(A) = 0$ implies linear dependence of columns of a matrix $A$ [closed]

Let $A$ be an $n \times n$ matrix. How would you rigorously prove that $\det(A) = 0$ if and only if the columns of $A$ are linearly dependent?
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1answer
54 views

finding projection on subspace

I have a question: Find the projection of $v = <1,2,1>$ on $span(<3,1,2>,<1,0,1>)$ in $R^3$ calling the vectors in the span a and b $$proj_w V = \frac{V \cdot a }{a^2} ...
6
votes
3answers
152 views

Finding the smallest sub-family of subsets needed to form a new subset

TL/DR I have a universe $U$ of items $u_i$ and a family $F$ of subsets of $U$ (call them $P_j$ ⊆ $U$). Given this family of subsets, I would like to find the sub-family $C$ ⊆ $F$ of subsets that can ...
1
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2answers
73 views

Angle between vectors?

Here's the problem from my homework: If the vector $\vec{a}+\vec{b}$ is perpendicular to the vector $7\vec{a}-5\vec{b}$, and if the vector $\vec{a}-4\vec{b}$ is perpendicular to the vector ...
3
votes
1answer
47 views

When solving via gauss-jordan

When I solve via Gauss-Jordan, taking a $3 \times3$ matrix as an example... Should I always try and get a $1$ in the upper left corner and $0$'s in the rest of the column followed by getting a $1$ in ...
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1answer
48 views

Does the identity matrix adapt to any other matrix?

So I have a matrix of the form $X=AX+B$ Where $X$ is a 3 by 1 column matrix, $A$ is a 3 by 3 matrix and $B$ is a 3 by 1 column matrix. (Notice that I am talking about Leontief input-output). So I ...
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2answers
85 views

characterization of uniform ellipticity

Let $B$ be a $n\times n$ matrix over $\mathbb{R}$ and define $A:=BB^*$. I read in a paper that the following two statements are equivalent: (1) the matrix $A$ is uniformly elliptic; i.e. for all ...
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1answer
87 views

vector subspaces of $(\mathbb Z/2\mathbb Z)^3$

How many possible vector subspaces of $(\mathbb Z/2\mathbb Z)^3$ are there? My idea was, to proove this as follow: $$U_b := ...
0
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1answer
222 views

Normal Operator that is not Self-Adjoint

I'm reading Sheldon Axler's "Linear Algebra Done Right", and I have a question about one of the examples he gives on page 130. Let $T$ be a linear operator on $F^2$ whose matrix (with respect to the ...
5
votes
2answers
108 views

Derivative of a map involving the matrix inverse

I have $f: U\rightarrow \mathbb{R}$, $f(X):=\operatorname{tr}(X^{-1})$, $U$ contains all matrices $X$, which are positive definite and symmetric. I want to show that $f$ is differentiable on $U$. To ...
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3answers
154 views

Dual Vector Spaces with Orthonormal Basis

I'm really stuck on the following quesiton. Let $U$ and $V$ be finite dimensional vector spaces over the complex numbers with bases $e_1,..,e_n$ of $U$ and $f_1,...,f_m$ of $V$. They also have dual ...
3
votes
3answers
61 views

Commutation of exponentials of matrices

Given two $n \times n$ real matrices $A$ and $B$, prove that the following are equivalent: (i) $\left[A,B\right]=0$ (ii) $\left[A,{\rm e}^{tB}\right] = 0,\quad$ $\forall\ t\ \in\ \mathbb{R}$ (iii) ...
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1answer
59 views

Easy question on reflection matrices

We have a reflection matrix $\textbf{A}\in\mathbb{R}^{n\times n}$ so $\textbf{A}^2=\textbf{I}_n$. I know that the possible eigenvalues of $\textbf{A}$ are $\pm1$. My question is: what can we say about ...
3
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0answers
95 views

Reflections in Dihedral Group

In Dihedral Groups, what is the meaning of reflection ? A line needs to be specified for a reflection to take place, but, if you specify only one line how will $D_n$ give all the symmetries for a ...
4
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2answers
136 views

How prove this $det\left(\frac{1}{\lambda^2_{i}+t\lambda_{i}\lambda_{j}+\lambda^2_{j}}\right)_{n\times n}>0,-2<t<2$

Question: Show that for $t\in (-2,2)$ and $0<\lambda_1<\lambda_2<\ldots<\lambda_n$ we have $$det(A)=det\left(\dfrac{1}{\lambda^2_{i}+t\lambda_{i}\lambda_{j}+\lambda^2_{j}}\right)_{n\times ...
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1answer
43 views

Proving linear independence.

Let $E$ be a $3$-dimensional vector space over the field of rational numbers. Suppose $T$ is a linear operator and $T(x)=y$, $T(y)=z$, $T(z)=x+y$ for certain $x$, $y$, $z$ in $E$ and $x\ne 0$. Prove ...
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1answer
62 views

Linear Transformation proof about one to one

Qu. $\textbf{4}\quad$ Suppose that $S$, $T:V\to V$ are linear transformations of a finite-dimensional vector space $V$, and suppose that the composition $ST:V\to V$ is invertible. Show that then ...
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0answers
39 views

Why the pivots and relative to the eigenvalue in symmetric matrix?

In the book, it said, there a quick fast way to test whether the eigenvalue are all positive or not. Just check the pivot of the symmetric matrix, if x no. of positive pivot, it would have x no.of ...
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0answers
40 views

A basic question on Jordan Canonical form and minimal polynomial

Suppose I am given a polynomial and asked to find out a linear map for which that is the minimal polynomial. Because we can read minimal polynomial from Jordan Canonical form, we can design matrix ...
6
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0answers
78 views

Does there exist an infinite $S \subset \Bbb R^3$ such that any three vectors in $S$ are linearly independent? [duplicate]

Does there exist an infinite subset $S \subset \Bbb R^3$ such that any three vectors in $S$ are linearly independent?
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2answers
95 views

How to show that exp is a diffeomorphism between symmetric reals and positiv definite matrices?

I am looking for an easy proof of the fact that the exponential function is a diffeomorphism between the finite dimensional vector space of symmetric real nxn-matrices and the open subset of positive ...
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0answers
20 views

A linear algebra problem on minimal polynomial

Suppose for a linear map $T$ in complex vector space $V$, $\lambda_1$ and $\lambda_2$ are the distinct eigenvalues such that $(T-\lambda_1 I)(T-\lambda_2 I)$=0. I need to prove that $ker(T-\lambda_1 ...
3
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1answer
117 views

Computational cost, power method and page rank

When solving the PageRank problem for $n$ web pages, it is necessary to find a solution of the eigenvector equation $$(fM)*p = p,$$ where $$fM = dM + (1 - d)Z$$ $$Z =\frac{1}{n}*ee^T$$ $$e =[1, 1, ...
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1answer
64 views

If $D$ is the operator of differentiation, prove $D^{2}$ is a self adjoint linear operator on V and find all its eigenvalues and eigenvectors

Suppose $V$ is the space of infinitely differentiable complex valued functions $f$ on $[0,\pi]$ such that $D^{2k+1}f(0) = 0 = D^{2k+1}f(\pi)$ for all integers $ k \geq 0$. Then V is a complex IPS with ...
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1answer
50 views

Householder matrix confusion

I read that: If $(I-2ww^T)x=y$ and $x \neq 0$ ($w^Tw=1$) then $$w= \dfrac{(x-y)}{\|x-y\|_2}.$$ I tested this for $x=[9,2,6]^T$ and $y=[-11,0,0]^T$ and it worked. But for some reason for $x=[1,2,3]^T$ ...
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1answer
558 views

Prove projection is self adjoint if and only if kernel and image are orthogonal complements

Let $V$ be an IPS and suppose $\pi : V \to V$ is a projection so that $V = U \oplus W$ (ie $ V = U + W$ and $U \cap W = \left\{0\right\}$) $ \ $ where $U = ker(\pi)$ and $W = im(\pi)$, and if $v = u ...
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2answers
54 views

Eigenvalues of a rank-one update to a rank-one matrix

Let $\mathbf{a}$ and $\mathbf{b}$ be two column vectors in $\mathbb{C}^N$. What can we say about the eigenvalues of the matrix \begin{align} \mathbf{a}\mathbf{a}^H+\mathbf{b}\mathbf{b}^H \end{align} ...
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1answer
78 views

Is there a simple method to finding orthonormal basis given a partially complete set

I have a question Find the indicated projection matrix for the given subspace, and find the projection of the indicated vector $<2,-1,3>$ on ...
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1answer
36 views

A question about the eigenvector and the basis

Let $(1, 0, 0)^T$ and $(0, -1, 1)^T$ be eigenvector of a 3x3 matrix $A$ with eigenvalue 1 and $(-2, -2, 1)^T$ be an eigenvector of $A$ with eigenvalue 2.Put $e_3=(0, 0, 1)^T$. Find eigenvector $v$ ...
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2answers
33 views

Matrix representaiton of linear operator by inner product

Let $V$ be a finite-dimensional inner product space, and let $\beta = \{\alpha_{1}, \cdots \alpha_{n}\}$ be an orthonormal basis for $V$. Let $T$ be a linear operator on $V$ and $A$ the matrix of $T$ ...
8
votes
3answers
420 views

Relationship between Nilpotent Matrix and Matrix with all zero diagonal factors.

solving Linear Algebra HW, I suddenly became curious about the relationship between Nilpotent Matrix and matrix with all zero diagonal factors such that $A_{11} = A_{22} = \cdots = A_{nn} = 0$ Does ...
3
votes
1answer
77 views

Transpose for an infinite dimensional vector space

Suppose that $V$ and $W$ are 2 finite dimensional vectors spaces and $T$ is a linear transformation such that $T : V \rightarrow W$. Then $T \rightarrow T^t$ can be seen as an isomorphism of $L(V,W)$ ...
2
votes
1answer
113 views

determinant of specific circulant matrices

I got problem in determining the determinant of specific circulant matrix $C$ formed by shifting the vector $1\cdots101\cdots10\cdots0$. The number of $1$'s in the first sequence of $1$'s is $k$ and ...
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4answers
191 views

$\vec{y}$ as the sum of two orthogonal vectors

I'm having difficulty getting this problem down. I have not idea what to do, and I can't find any leads from my notes. Any advice? Let $\vec{y}$ = $ \left[ {\begin{array}{cc} 3 \\ -5 ...
3
votes
2answers
94 views

How find this matrix value of this $\det(A_{ij})$

Find this value $$\det(A_{n\times n})=\begin{vmatrix} 0&a_{1}+a_{2}&a_{1}+a_{3}&\cdots&a_{1}+a_{n}\\ a_{2}+a_{1}&0&a_{2}+a_{3}&\cdots&a_{2}+a_{n}\\ ...
0
votes
3answers
248 views

Computing the Distance

I'm having a little difficulty getting this problem down. I've been trying to follow my notes, but I guess I'm not doing it correctly. Anyone know how to properly answer this question? Let ...
0
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3answers
66 views

Proof of Linear Independence of Vectors after Applying a Linear Transformation

Does any know how to go about proving the following statement? Let $v_1, v_2, \dots, v_n \in V$ be a linearly independent vectors. Furthermore, let $T \in \mathcal{L}(V)$ be an invertible linear ...
0
votes
2answers
153 views

How to tell if a map is a linear map?

Can someone run me through the process of showing whether a map is a linear mapping or not. For an example I have: $T:\mathbb{R}^2 \to \mathbb{R}^2, T(x,y)=(x-y^2, 5x)$ I am aware that it must ...
0
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1answer
60 views

diagonalizability and invariant subspaces [duplicate]

My question is about linear algebra, especially invariant subspaces and diagonalizability. Here is the question: Let $A$ be a diagonalizable linear operator on the finite dimensional vector space ...
0
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0answers
56 views

Linear Algebra Help - Eigenvalues and Convergence

If A is a real symmetric matrix, how do the signs of the eigenvalues of A affect converge of Bk, where Bk is defined below. Bk = A^k / trace(A^k), where A^k is the kth matrix power of A. When all ...
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1answer
32 views

projections of a vector space and a linear operator on this vector space

i have a question about linear algebra,especially about invariant subspaces and projections.the question is below: let P be a projection of the vector space V over a field F and let T be a linear ...
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4answers
68 views

Finding $A^k$ for non-diagonalizable $A$

Is there an easy way to find $A^k$ for a square matrix $A$ that is NOT diagonalizable?
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2answers
73 views

Show that the matrix is invertible

let $A \in M_n(F)$ be a n by n matrix with values from an unknown field $F$. $P_A(t)$ is the characteristic polynomial of $A$, and $g(t) \in F[t]$ a polynomial of an unknown degree. assume that ...
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1answer
39 views

Does every real vectorspace have a symetric positive definite bilinear form?

Does every real vectorspace $V$ (possibly not finite dimensional) have a symetric positive definite bilinear form? That is a map $s:V \times V \rightarrow \mathbb{R}$ such that: $$\forall v, w \in ...
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1answer
61 views

Question regarding $A = B^{-1}DB$ and determinants

Consider $A = B^{-1}DB$, where $A$ is a normal matrix represented by unitary matrices $B, B^{-1}$ and the diagonal matrix $D$. Although $B^{-1}B = BB^{-1} = I_B$ why doesn't $B^{-1}DB$ give you $D$? ...
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3answers
48 views

How to prove Linear Independence

How to prove the set $S=\{x,|x|\}$ is linearly independent. Where S is a subset of set of real valued functions on $\mathbb{R}$. Thank You.
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205 views

Finding eigenvalues of an uknown matrix subtracted by the identity

The question If the eigenvalues of A are 0, 1, and 3, find the eigenvalues of A-I. Explain how you obtained them. My intuition is telling me that I just subtract ...
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2answers
64 views

Unitary map between sets of vectors

Suppose I have two sets of vectors, $E_1=\{v_i\}_{i=1}^{k}$ and $E_2=\{u_i\}_{i=1}^{k}$, with each vector belonging to $\mathbb{C}^k$. When is it possible to find a unitary matrix that maps $E_1$ to ...
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1answer
36 views

On finding an expression of this matrix.

Let $M_n$ be a $n × n$ matrix with real coefficients of which the entry in the $i$-th row and the $j$-th column equals 1 whenever $|i − j| ≤ 1$ and 0 otherwise. Is it possible to find a general ...
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4answers
145 views

Linear Algebra and Set Theory book recommendations.

I would like to studying linear algebra and set theory. Does anyone have a a good recommendation of books/resources/etc.?