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 ...

learn more… | top users | synonyms

2
votes
1answer
13 views

Number of possible rectangles from at most N identical squares

I was looking to find the number of distinct rectangles possible from at most $N$ identical squares. (Two rectangles are distinct if one cannot be rotated to obtain another) e.g for $N = 6$ , $8$ ...
0
votes
0answers
18 views

When a system of rational linear homogeneous equations have complex solutions

Question: When a finite system of rational linear homogeneous equations in finitely many variables have a nontrivial complex solution (that is not a rational solution), does it imply that there is ...
0
votes
1answer
11 views

Prove another matrix is positive definite given that A is a Hermitian matrix

Suppose that $A$ is a Hermitian symmetric $n\times n$ matrix of complex numbers all of whose eigenvalues lie inside the interval $(-1,1).$ Prove that the matrix $A^3+Id$ is positive definite. An ...
0
votes
1answer
26 views

Norms are continuous maps [on hold]

How can I prove that the mapping $$ f:\mathbb{R}^{N}\rightarrow\mathbb{R}, $$ defined by $$ f(\mathbf{x})=\|\Phi\mathbf{x}\|^{2}_{2}, $$ is continuous?
2
votes
1answer
12 views

Weighted average of multiple points

Let's say I have a triangle whose three corners are $$(x_1,y_1),(x_2,y_2),(x_3,y_3).$$ I have a weight assigned to each one as a percentage, so the first point might be $75\%$, the second $15\%$ and ...
0
votes
2answers
37 views

How do I find basis for a polynomial ?How should I start for question like those?

Question such like, (1) Let $V = P^4$ be the vector space of all real valued polynomials of degree less than or equal to four. Let $W =\{p(x)∈P^3 |p(−2)=p(2)\}$. Find the basis for $W$ (2) Let $U = ...
-2
votes
1answer
25 views

Question about intersections [on hold]

"Consider the intersection of the functions $y=\frac{m}{10x} + m$ and $y=\frac{m}{x}$ where $m$ is a real number. Investigate the values of m that may provide, 0, 1, or 2 points of intersection." ...
0
votes
1answer
22 views

Change of basis from falling powers to powers for polynomials up to degree $n$

Notice that $$(1, x, x^{\underline{2}}, x^{\underline{3}}, \dots)$$ and $$(1, x, x^2, x^3, \dots) $$ both are bases of $\mathbb{R}[x]$ (where $x^{\underline{n}}$ is the falling power). Now suppose the ...
1
vote
1answer
19 views

How could I calculate the local size of an object given its distance and actual size?

Lets say I take a picture of a sign. I know that sign is 20ft (width), 10ft height. I'm standing 40 feet away. If I were to take a picture, how could I calculate how wide and how high the sign is in ...
6
votes
3answers
171 views

Solving recurrence relation: Product form

Please help in finding the solution of this recursion. $$f(n)=\frac{f(n-1) \cdot f(n-2)}{n},$$ where $ f(1)=1$ and $f(2)=2$.
0
votes
0answers
25 views

a proof question regarding to eigenvalues and diagonalization

Let the scalar field be $\mathbb{F}$. Let $T: V\rightarrow V$ be a linear operator represented by the $n\times n$ matrix $A= [T]_{\alpha\alpha}$. Suppose that the characteristic polynomial of $A$ ...
2
votes
1answer
40 views

Eliminate 2 variables from 3 equations with lots of parameters

I want to eliminate the variables x and y from these 3 equations in a way that all parameters appear in one equation without x and y: ...
0
votes
1answer
30 views

Determine whether the following map is a linear transformation.

So I have to determine if the following is a linear transformation: $$T: F(I) \rightarrow F(I)$$ defined by: $$T(f) = 2f$$ I know that if you let $T: V\rightarrow W$ be a linear transformation. Then: ...
1
vote
1answer
28 views

Projection equation

I'm a programmer, not a math expert or statistician by any means, but my organization wants a page in our admin console that displays a projection of how many registrations we can expect to see based ...
0
votes
2answers
22 views

Largest eigenvalue of a block diagonal matrix is an eigenvalue of the largest block?

Consider this square matrix $C = \begin{bmatrix} A& 0 \\ 0 &B \end{bmatrix}$, where $A$ and $B$ are also square matrices. Suppose $A$ is larger in the sense that is an $n \times n$ matrix, ...
0
votes
1answer
30 views

Isometry in a finite dimensional vector space is always surjective

My book defines an isometry as a linear operator between two vector spaces X and Y where: $$\|T(x)\|=\|x\|$$ Later it has a sentence which I do not understand. If we have a finite dimensional ...
0
votes
4answers
30 views

Surjective function - proving

$f: \mathbb{R}\to \mathbb{R}$ $f(x) = x^3 -2x^4$ In order to prove that $f$ is not surjective, my teacher told me to find that in most the $f$ is negative. And indeed, only for $0<x<0.5$ it's ...
1
vote
0answers
19 views

How to find the rank of a toeplitz matrix?

Is there any trick to compute or estimate the rank of a toeplitz matrix ? Or is this still unknown for a general toeplitz matrix ?
5
votes
1answer
52 views

Determinant of the linear map given by conjugation.

Let $S$ denote the space of skew-symmetric $n\times n$ real matrices, where every element $A\in S$ satisfies $A^T+A = 0$. Let $M$ denote an orthogonal $n\times n$ matrix, and $L_M$ denotes the ...
1
vote
1answer
27 views

Surjective functions and cal'

$f,g: \mathbb{R}\to \mathbb{R}$ Both are also surjective functions. My question is if $f+g$ will be also surjective. I need to dis/prove it if it's true or false. Now, my friend told me it's false ...
1
vote
0answers
19 views

Is the steinitz exchange lemma necessary to establish invariance of 'basis-size'?

I am going to answer my own question in some sense... In Beardon's "Algebra and Geometry" he proves (Theorem 7.2.2) that if $v_1,\ldots,v_n$ and $u_1,\ldots,u_m$ are both bases for some $F$-vector ...
3
votes
1answer
57 views

Is it true that $u + v$ is an eigenvector corresponding to the eigenvalue $\lambda$?

Let $A$ be an $n \times n$ matrix, and $u, v$ be eigenvectors corresponding to an eigenvalue $\lambda$ of $ A$ (that is, $Au = \lambda u$ and $Av = \lambda v$). Is it true that $u + v$ is an ...
0
votes
0answers
27 views

Tree Traversal - Simple Puzzle type Issue.

This is a puzzle like question,based on Fibonacci like structure of the tree. Actually it is a short question with out any complex concepts. It appears bit big,since I have added explanations with ...
1
vote
1answer
26 views

Diagonal matrices and integrals

Suppose that $$A=\int_{\alpha}^{\beta} f(B,x)\ dx,$$ where $B$ is a $3\times3$ matrix. The result I'm looking for is that if $B$ is diagonalized with an orthogonal matrix, then is A diagonalized by ...
0
votes
1answer
13 views

Minimal polynomial of a matrix whose elements have a certain form

Find the minimal polynomial of the $n$-dimensional matrix $(a_{ij})$ when the matrix elements $a_{ij}$ have the form $a_{ij} = u_i v_j.$ Let $A=uv^T$ where $u,v$ are column vectors. Then ...
1
vote
1answer
121 views

Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0 $$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
0
votes
1answer
13 views

LLL algorithm in pari/gp

I know that in PARI/GP the function qflll performs LLL algorithm on a set of bases. However, is it possible for me to look at the code for ...
3
votes
2answers
39 views

Pairwise commuting nilpotent matrices: alternative solution needed

I have a problem: Let $A_1,A_2,...,A_n$ be $n\times n$ nilpotent matrices which are commute in each pair ($A_iA_j=A_jA_i$). Prove that: $$A_1A_2...A_n=0$$ I have got a solution by proving that ...
1
vote
1answer
56 views

Maximum determinant of a $m\times m$ - matrix with entries $1..n$

I want to find the maximal possible determinant of a $ m\times m$ - matrix A with entries $1..n$. Conjecture 1 : The maximum possible determinant can be achieved by a matrix only ...
1
vote
2answers
37 views

Fourier coefficients with respect to an orthonormal basis for an inner product space

$V = \operatorname{span}(S)$, where $S = \{(1, i, 0), (1 - i, 2, 4i)\}$, and $x = (3 + i, 4i, -4)$. Apply the Gram–Schmidt process to the given subset $S$ of the inner product space $V$ ...
0
votes
0answers
21 views

Calculating dimension of subspace of all commutating matrices with fixed matrix A . [duplicate]

$A$ is a given point in the vector space $M_{n\times n}(F)$ on the filed F. define the subspace $$W_A=\{B\in M_{n\times n}(F) : AB=BA\}$$ of $M_{n\times n}(F)$ . The question is this that what is ...
0
votes
2answers
48 views

Characteristic polynomial and eigenvalues of a $3 \times3$ matrix.

Hi so I have to find the characteristic polynomials and the eigenvalues of the matrix: $$A = \begin{bmatrix}1 & 0 & 3\\2 & -2 & 2\\3 & 0 & 1\end{bmatrix}$$ So I know you use ...
0
votes
1answer
25 views

Determining if Linear transformation

Please help me get and understand this concept of linear algebra based on this questions: Determine whether or not $T$ is a linear transformation from $\Re^2$ to $\Re^2$ if u$ \in \Re^2 $ and v$ \in ...
4
votes
2answers
57 views

Rewriting the matrix equation $AX = YB$ as $Y = CX$?

Is it possible in general, if $A,B,C,X,Y$ are square and of the same dimensions? If so, does it generalize to non-square matrices (using a pseudoinverse)? I'm doing some curve fitting in which I have ...
0
votes
1answer
64 views

Characterize matrices A such that trace(AC)=0 for every matrix C with trace(C)=0

$A$ is an $n\times n$ matrix on the field $F$ such that for every $n\times n$ matrix $C$ with $\operatorname{trace}(C)=0$ we have $\operatorname{trace}(AC)=0$. Can we characterize such matrices $A $? ...
0
votes
0answers
34 views

Bilinear form equations [on hold]

$\mathit{V}$ is 3-dimensional vector space over complex. Complex valued square matrix $R_A, R_B, R_C\in M_3(\mathbb{C})$ are given. Suppose that both $R_A, R_B$ and $R_C$ are Hermitian ...
0
votes
0answers
22 views

Which casses of matrices contain A and which contain B? Linear Algebra

Am pretty confused about classes, I don't know what it means, so so I can't really do part_A and I need your help with it? For part B, I got all eigen = 1 for matrix A, and 0 for matrix B, Is this ...
2
votes
2answers
25 views

The inverse of a linear transformation $A$ can be expressed as a polynomial in $A$

Suppose that $A$ is a non-singular linear transformation of an $n$-dimensional linear space into itself. Show that there exists some polynomial $c_0+c_1z+\ldots+c_kz^k$ so that ...
0
votes
1answer
43 views

Dimension of $\dim_{\mathbb C}\big(\mathbb C[X,Y]/I(Y^2-X^2,Y^2+X^2)\big)$

I'm trying to solve the question 1.36 from Fulton's algebraic curves book: Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and $\dim_{\mathbb C}(\mathbb C[X,Y]/I)$ Obviously ...
2
votes
1answer
39 views

Method of orthogonalization that preserves invertibility

Is there a method of orthogonalization such that, given an invertible matrix $A$ with entries in the real numbers, applying the method and then inverting the result is the same thing as applying the ...
0
votes
0answers
32 views

What is the linear combination of B?

I have a problem where I am finding $A^n$B where B=$[3,1,1]^t$. I know the steps in solving, but I do not remember how to find linear combination. I do not see it. There has to be a way to calculate ...
1
vote
0answers
10 views

Trouble understanding Hoffman / Kunze exercise [duplicate]

I am finishing up a number theory class and will be studying graduate Linear Algebra in the fall so I thought I'd start early getting familiar with the text and authors by doing some of the early text ...
0
votes
1answer
18 views

How do you get nullspace N(A) to be orthogonal to C(A^H)

In the picture below, C(A) is given in number7, but I am doing number_8. Ii did a gauss jordan where by i subtracted R2-iR1 to get 0 belo 1st pivot and 1 as the second pivot in column2, row2. Then I ...
1
vote
1answer
20 views

Particular solution to the matrix form Ax=b

This question is more for general understanding than looking for a specific answer. I have a theorem that states: "If Ax=b has a solution x$_p$, then the general solution to the equation is x$_p$ + ...
3
votes
1answer
60 views

Coproduct in the category of vector spaces with bilinear forms

I'm trying to work out the coproduct in the category of (say real) vector spaces equipped with bilinear forms, where the morphisms $(V,b) \to (V',b')$ are the linear maps $T : V \to V'$ such that $T^* ...
1
vote
1answer
25 views

Prove existence of Diagonalizable Matrix

Suppose R, T $\in L(F^3)$ each have 2, 6, 7 as eigenvalues. Prove that there exists an invertible operator S $\in L(F^3)$ such that $R=S^{-1}TS$. What I got so far is that since R and T have three ...
1
vote
2answers
36 views

Why use Gauss Jordan Elimination instead of Gaussian Elimination, Differences

Why use Gaussian Elimination instead of Gauss Jordan Elimination and vice versa for solving systems of linear equations? What are the differences, benefits of each, etc.? I've just been solving ...
0
votes
3answers
65 views

Orthogonal diagonalization of Symmetic Matrices

Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial \Delta (t). Step 2: find the eigenvalues of A which are the roots of \Delta (t). Step 3: for each ...
1
vote
3answers
67 views

Proof that $\mathrm{ker}(A^{T}A) = \mathrm{ker}(A)$?

Is there a proof that can help me understand why this is the case? I can't conceptualize the reason for this in my mind. Thanks.
0
votes
1answer
44 views

Eigenvectors for normal operators and their adjoints

Can someone tell me if this proof is correct? Claim:V is a vector space over the Complex field. $T:V\rightarrow V$ is a normal operator. Then if $v\in V$ is an eigenvector with the eigenvalue ...