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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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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45 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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69 views

Orthogonal projection matrix [closed]

Let $P\colon \mathbb{R}^3\to\mathbb{R}^3$ be the orthogonal projection on the line $L$, where $L = \operatorname{span}\{(-1,2,5)\}$. Determine the standard matrix of $P$. Does anybody know how to ...
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427 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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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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57 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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34 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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31 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 $T$ ...
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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 ...
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1answer
61 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 $L(V,W)$ ...
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1answer
104 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
136 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 ...
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2answers
90 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}\\ ...
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3answers
152 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 ...
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3answers
47 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 ...
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2answers
70 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 ...
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1answer
56 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 ...
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0answers
51 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
26 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
67 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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64 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
36 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
60 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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42 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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108 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
49 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
35 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
128 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.?
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48 views

Do $X(X'X)^{-1}(X'X)^{-1}X'$ and $(X'X)^{-1}$ have the same non-zero eigenvalues?

Let $X$ be a matrix with full column rank. I want to show that $X(X'X)^{-1}(X'X)^{-1}X$ and $(X'X)^{-1}$ have the same non-zero eigenvalues. I have checked it in matlab and Stata and the result holds ...
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Error of the norm of solution in linear least-squares

How can we estimate the solution norm ($\Vert x \Vert$) error, separate from the solution ($x$) error in solving $Ax=y$ (linear least-squares problem)? Is the error of $\Vert x \Vert$ higher or lower ...
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2answers
154 views

How prove this two symmetric matrices $AB=0$

Let $A,B$ be real symmetric matrices, and for any $n\in \Bbb N^{+}$, and for all $x,y\in \Bbb R$, we have $$tr(xA+yB)^n=x^ntr(A^n)+y^ntr(B^n).$$ Show that $AB=0$. My try:since ...
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1answer
53 views

the rank of matrix products including a commutation matrix

Given a full rank matrix $A \in \mathbb{R}^{M \times N^2}$ where the rank of ${A}$ is ${\rm rk}(A)= M \leq N^2$ and the commutation matrix $K_{NN}$. I need to find the rank of a matrix product ...
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2answers
73 views

Matrix of transform rotation [solved]

Im trying to create matrix which rotates vector. I have $\vec{g}=(g_1,g_2,g_3);\:g_1\in\mathbb{R},g_2\in\mathbb{R},g_3\in\mathbb{R}$ - it represents gravitation. And $\vec{o}=(o_1,o_2,o_3)$ is vector ...
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1answer
163 views

Generalization of $\frac{a + b}{c + d} \leq \text{max}(\frac{a}{c}, \frac{b}{d})$

I'm looking for a matrix version of the basic inequality for the ratio of two sums of positive numbers: $$\frac{a + b}{c + d} \leq \max\left\{\frac{a}{c}, \frac{b}{d}\right\}.$$ Specifically, I have ...
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6answers
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The milk sharing problem

I found a book with math quizzes. It was my father's when he was young. I encountered a problem with the following quiz. I solved it, but I wonder, is there a faster way to do it? If so, how can I ...
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Orthgonal complement of $U=\{X\in M_2(\mathbb R): X=\left(\begin{array}{cc} a+2b&a-b\\ b&a \end{array}\right), a, b\in\mathbb R\}?$

Let $M_2(\mathbb R)$ endowed with the inner product $\langle A, B\rangle=\textrm{tr}(B^TA)$. What would be an easy way to find the orthogonal complement of $$U=\{X\in M_2(\mathbb R): ...
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2answers
220 views

non-degenerate symmetric bilinear form, show $\dim F+\dim F^{\perp}=\dim E=\dim\left(F+F^{\perp}\right)+\dim\left(F\cap F^{\perp}\right) $

Lang uses this formula in proposition 1.2, page 573 of his book "Algebra" (Graduate). I did not find a straight forward argument to show it, the only argument I found was really long, and not very ...
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2answers
75 views

Proof that $(\alpha I - A)$ invertible if $\alpha > \rho(A)$

I want to proof that for $A \in \mathbb{R}^{n \times n}$ with $a_{ij}\geq 0, \forall i,j=1,...,n$: \begin{align} (\alpha I - A) \text{ is invertible if } \alpha > \rho(A) \end{align} where ...
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1answer
41 views

Linear spans are equal if they contain one another's generators

Let $V$ be a vector space, and suppose $M = [\{a_1, a_2, \ldots, a_k\} ]$ and $L = [ \{b_1, b_2, \ldots, b_m\} ]$ are subspaces of $V$. Prove that $M = L$ iff $a_i$ is an element of $L$, for every ...
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1answer
44 views

Linear functional and transposition. [duplicate]

This is a question from Hoffman. Let $V$ be a finite-dimensional vector space over the field $F$ and let $T$ be a linear operator on $V$. Let $c$ be a scalar and suppose there is a non-zero vector ...
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2answers
107 views

Is integral operator surjective?

Is the linear operator ${\rm A}$ given by $${\rm A}f(t) = \int_0^t f(x)\ dx$$ onto? It would be onto if the range space of ${\rm A}$ is equal to whatever ${\rm A}$ is mapping to, but I am not even ...
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1answer
36 views

Find all integer for equation

I have the following equation. I need to find all integers. $241 \times a + 114 \times b = 1$ As per Wolframalpha the solution would be: $a=114n+79, b=-241n-167,n \in \mathbb{Z}$ But how do you ...
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1answer
45 views

If dim(V) + dim(W) > dim(R^n), show that some nonzero vector is in V & W. [GStrang P183, 3.5.45]

Inside $\mathbb{R^n}$, suppose dimension$(\mathbf{V})$ + dimension$(\mathbf{W}) > n$. Show that some nonzero vector is in both $\mathbf{V}$ and $\mathbf{W}$. Answer : Since $\dim\mathbf{V} ...
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2answers
128 views

Isometric isomorphism between infinite sequences and $L^2[-1,1]$

Show that the set of all infinite sequences $(x_1, x_2, ...)$ where $$\sum\limits_{n=1}^\infty x_n ^2 < \infty$$ is isometrically isomorphic to $L^2[-1,1]$.
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132 views

Orthogonal Complements and Subspaces Proof

I'm having a little difficulty understanding the proof for orthogonal complements. I kind of understand orthogonal complements, but I cannot seem to find a logic to this. I'm trying to follow along ...
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3answers
50 views

Has this system unique solutions?

Can i find solutions for this system of equations? $x\cdot y= a$ $\frac{x}{y}= b$ in which x,y are unknown and a,b known values? My only hesitation is that the system has many solutions and the ...
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2answers
231 views

Transpose of an invertible linear transformation..

I am trying to prove that suppose that a linear transformation $T$ is invertible, then its transpose $T^t$ is also invertible. Is the following proof correct? Proof: Let $T$ be an invertible ...
2
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1answer
70 views

Something about Gram-Schmidt Projections

Recently I'm reading the book Numerical Linear Algebra and I have a problem in Lecture 8, Gram-Schmidt Orthogonalization. The following text is from the book. Let $A\in\mathbb{C}^{m\times n}$, ...
3
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1answer
96 views

Does 1 distinct eigenvalue guarantee 1 eigenvector?

I am trying to figure out when 2x2 matrices are not diagonalizable. Right now, my conditions are: the matrix has only 1 distinct eigenvalue the matrix yields only 1 linearly independent eigenvector ...
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0answers
57 views

Inward-pointing normal and co-ordinate systems

I'm doing a course in computer graphics, and as such, we're being taught measures on how to deal with the Hidden Surface Removal problem. One of the topics covered was "back-face detection", that is, ...