# Tagged Questions

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 ...

47 views

### How to compute this determinant, without the Sylvester determinant theorem, [duplicate]

The problem statement is: Show that there exists numbers $a$ and $b$ such that $$det (A + sxy^*)= a+bs$$ here $A$ is an $nxn$ matrix with real entries, and $x,y\in R^n$. I've been using brute ...
24 views

### Find a direct basis for boundary orientation

Let M in R4 be a manifold defined by equation d= $x^2$ + $y^2$ +$z^2$ and oriented by sgn $dx_1$^$dx_2$^$dx_3$. Consider the subset where d<=1. Show that it is a piece with boundary. Let x be a ...
38 views

### Prove that dim$($Ker$(A^n))$ is odd for all $n$ sufficiently large

I have the following problem: Let $A$ be a matrix of $101$ by $101$ over $\mathbb{F}_2$, such that trace$(A^n)=0$ for all $n$ positive integer. Prove that dim$($Ker$(A^n))$ is odd for all $n$ ...
40 views

### Proof about Diagonalization of A

The question asks WHY is it true that $$A^{n} = PD^{n}P^{-1}$$ I can never do proper proving in algebra; what I almost know for sure is that a proof by induction is the way to go here. But how do you ...
44 views

### Prove that p divides to algebraic multiplicity of the eigenvalue

I need help in the following exercise of a qualifying exam: Let $A$ be a matrix of size $m$ by $m$ over the finite field $\mathbb{F}_p$ such that $\operatorname{trace}\left(A^n\right)=0$ for all $n$. ...
41 views

### QR Factorization for Inconsistent Linear System

I am trying to recreate the problem found here on finding the least squares solution to an inconsistent linear system via QR factorization. Can someone explain the part about adding on vectors so that ...
31 views

### make two sets of eigenvectors orthogonal

Suppose that $v_1,v_2,v_3$ are orthogonal eigenvectors of $A$ and $u_1,u_2$ are orthogonal eigenvectors of $B$. Is there any way to choose $v_1,v_2,v_3$ and $u_1,u_2$ such that $v_1,v_2,v_3,u_1,u_2$ ...
43 views

### Prove rayleigh quotient = operator norm without referring to eigenvalues

Let $H$ be a Hilbert and $T \in \mathcal{L}(H,H)$ a symmetric operator. Prove $$\|T\| = \sup\{|(x,Tx)| : x \in H, \|x\| = 1\}$$ without referring to the eigenvalues of $T$ (which is what all ...
29 views

### Projection of Range of Matrix $A$ onto Kernel of $A-I$

Let A be a $2 \times 2$ matrix such that $A^2 = A$. Show that $Ax = x$ for every $x$ in $R(A)$ and if $rank(A)$ $= 1$, $M = R(A)$, $N = Ker(A-I)$ then $A$ is the projection along $N$ onto $M$. For ...
136 views

### Measure theory , Functional calculus, Self Adoint

In $$L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. I need to show $\langle g_a|g_b \rangle=0$ if ...
34 views

### There exists an algebraic basis $(e_i)_{i \in I}$ in $E$ such that $\|e_i\| = 1$ for all $i \in I$?

Let $E$ be a infinite-dimensional normed vector space. How do I see that there exists an algebraic basis $(e_i)_{i \in I}$ in $E$ such that $\|e_i\| = 1$ for all $i \in I$?
688 views

46 views

104 views

49 views

### Minimal polynomial of a matrix of matrices.

Let the minimal polynomial of a matrix A be equal to $f(x)=(x-c_1)^{n_1}…(x-c_k)^{n_k}$. Prove that the minimal polynomial of the matrix $\begin{pmatrix}A & I \\0 & A \\\end{pmatrix}$ is equal ...
21 views

### How to show that the functionals given form a basis for the dual space.

I've done part $(a)$, but not sure about part $(b)$. If I understand the quetion correctly, then if $f$ is just a general polynomial of degree less than or equal to $n$, then $E_{x_0},\ldots,E_{x_n}$ ...
17 views

### Diagonalizability of a Linear Map

Please could somebody verify this: $T$ is diagonalizable $\iff$ there exists a basis of $V$ (the vector space ) consisting of eigenvectors of $T$ $\iff$ the algebraic multiplicity of each eigenvalue ...