# linear algebra

1. ### Calculate a determinant of specific a matrix

k, l, m, a0, a1, a2 are real numbers and sequence $\{W_n\}_{n=0}^\infty$ is determined in this way: W0 = a0, W1 = a1, W2 = a2, Wn = k*W(n-1) + l*W(n-2) + m*W(n-3) for n > 2. For n > 1, how to calculate: \[ \begin{vmatrix} \begin{pmatrix} W_{n+2} & W_{n+1} &W_{n}\\ W_{n+1}...
2. ### Using the spectrum to calculate edges and triangles

So I found out that $\sigma(K_6)=\{[-1]^5,\}.$ However how would I use this to calculate the number of edges and triangles in K6?
3. ### Minors order 2 in a square matrix

How many minors of order 2 can I extract from a generic square matrix?

5. ### Linear isomorphism

How do I show the below is a linear transformations . I’ve shown it’s 1-1
6. ### Linear transformations

Let A={ex,sin(x),excos(x),sin(x),cos(x)}A={ex,sin⁡(x),excos⁡(x),sin⁡(x),cos⁡(x)} and let V be the subspace of C(R)C(R) equal to span(A)span(A). Define T:V→V, f↦df/dx. T:V→V,f↦df/dx. How do I prove that TT is a linear transformation? (I can do this with numbers but the trig is throwing me).
7. ### a simple system with an impossible answer...

first post here... currently an applied math student attending Stonybrook university. The question seems so simple. But I am unable to find an answer. jack can do 3 chemistry problems and 6 math problems an hour. Jill can do 4 chemistry problems and 7 math problems an hour. How long must...
8. ### The volleyball net has the shape of a rectangle measuring 40 by 2017 cells. What is t

The volleyball net has the shape of a rectangle measuring 40 by 2017 cells. What is the biggest number of ropes can be cut so that the grid does not fall into pieces?
9. ### determinant of symmetric matrix of size (n+1)(n+1)

Let for j = 0,. . . n aj = a0 + jd, where a0, d are fixed real numbers. Calculate the determinant of the matrix A of size (n + 1) Ã— (n + 1)
10. ### proof of property

Let a 3 Ã— 3 matrix A be such that for any vector of a column v âˆˆ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.
11. ### Sudoculus

Hi everyone, I am a software engineer by profession, and a math geek by hobby. I have been working on a new game concept that combine my two passions in a fun way. The prototype for the game could be found at https://www.sudoculus.com, The game is based on the sudoku puzzles, and...
12. ### Linear system

Question: Using the fact that \textbf{Ax=b} is consistent if and only if \textbf{b} is a linear combination of the columns of \textbf{A} to find a solution to \left( \begin{array}{cccc} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \\ 3 & 4 & 1 & 2 \end{array} \right)\left( \begin{array}{c} x\\ y\\...
13. ### Reciprocal of x/y - 1

Which of the following equals the reciprocal of x/y - 1, where x/y - 1 does not equal zero? (A) 1/x - y (B) -y/x (C) y/(x - 1) (D) x/(xy - 1) (E) y/(xy - 1) My answer is x/y - 1 = (x - y)/y and its reciprocal is y/(x - y) but I am not able to get the right answer from the choices. Please help.

Considering the two basis of R. The 1st , B is the canonical base i,j defined by i={1 0} (i is column) and j={0 1} (j is column). The 2nd, BÂ´ is associated to matrix [P]=[1 -1; 1 1] (1 -1) ***(1 -1) is first column and (1 1) is second column. A vector V is expressed as V= {1 2} ***(1 2)...

How to prove this property of adjoint? I checked it on few examples, but I don't know how to prove it rigorously.
16. ### Dimension of the subspace T of HomV

L<V is a subspace of vector space V. T is a set of all linear maps from HomV (HomV={f : V->V, f is linear map}) for which the restriction f|L is a zero map. What is the dimension of T? I tried to find the range of f, but I'm obviously doing it wrong. :furious: Thank you and sorry for my English. :)
17. ### Modifying Solution of System of Linear Equation

Consider that we have a linear system of equations resulting in the following matrix equation $Ax=b$ where $A$ is a $3\times 3$ matrix, $x$ and $b$ are $3\times 1$ vectors. Let's assume that we know the solution of this system of equation and represent it with $y$ whose entries are given by...
18. ### f(Ï†(x))=Ï†(2)Ï‡

f(Ï†(x))=Ï†(2)x I saw an excercise with this weird looking linear transformation and I would like some help to find its matrix. f(g(x))=g(2)x Here's my thought: So let B=\left \{ u_{1},u_{2},u_{3}\right \}=\left \{ (x^2+x+1),(x),(1)\right \} be a basis of \mathbb{R}_{2}[x] then...
19. ### Prove the identity

Prove the identity Au.v = u.A^T v Note that "." in the equation means dot product. I know that I should write the dot products as products of matrices but I don't know how to do it. Thanks in advance :)
20. ### Orthonormal collection

Why any orthonormal collection of vectors is linearly independent?