# Tagged Questions

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### $A$ is an $n \times n$ invertible matrix, prove that $f(\mathbf u, \mathbf v)= \mathbf u^TAA^T \mathbf v$ defines an inner product on $\mathbb R^n$

I have difficulty especially proving that $f(\mathbf v, \mathbf v) \geq 0$ for all $\mathbf v$. Thanks
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### Distance of a point to a plane

Let $T$ be the plane $x+2y+3z=11$. Find the shortest distance $d$ from the point $P=(2, 4, 5)$ to $T$, and the point $Q$ in $T$ that is closest to $P$. This is just one of the questions on my ...
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### Show $\langle , \rangle |_W$ non degenerate $\implies$ $\langle , \rangle |_{W^\perp}$ non degenerate

Let $W \subset V$ be a subspace and $\dim V < \infty$. If $\langle , \rangle$ and the restriction $\langle , \rangle |_W$ are non degenerate, then $\langle , \rangle |_{W^\perp}$ is non degenerate ...
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### Systems of Linear Differential Equations - Is this Correct?

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
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### Linear map over a vector space of polynomials

Let $F$ be a field and Let $F_{n+1} [X]$ (odd notation, in my opinion) be the vector space of polynomials of degree less than or equal to $n$ over $F$. Define $t: F_{n+1}[X] \to F_{n+1}[X]$ by ...
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### Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
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### What are the Routh Hurwtiz Criteria for 3$\times$3 Matrices?

The Criteria I know (for dynamical systems) is... The eigenvalues of a matrix are guaranteed to be negative if Tr($J$)<0 and det($J$)>0, where $J$ is the Jacobian of some 2 dimensional dynamical ...
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### Calculate Matrix A from eigenvalues, but no given eigenvectors

Here is my question: Write down a nontriangular 3 by 3 matrix whose eigenvalues are 6, 9, 2. I understand that you can calulate Matrix A using the formula A=V$\Lambda$$V^-1$, but is there a way to ...
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### For a shift matrix $A$, prove that $A^n=0$ but $A^{n-1} \neq 0$.

Let $A\in F_n$ be the matrix $\begin{pmatrix} 0&1&0&0&\cdots&0 \\ 0&0&1&0&\cdots&0 \\ \vdots\\ 0&0&0&0&\cdots&0 \end{pmatrix}$, whose ...
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### Definition: Eigenvalues of a matrix

1) Can a non-square matrix have eigenvalues? Why? 2) True or false: If the characteristic polynomial of a matrix A is p($\lambda$)=$\lambda$^2+1, then A is invertible. Thank you!
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### Trouble understanding finite vector spaces and Gaussian coefficent

I have studied linear algebra for 2 months now and i cannot understand a task that i am currently trying to solve. Basically i am trying to find the amount of bases for n-dimensional vector space over ...
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### Matrix Transformation - Using matrix multiplication

How do I use matrix multiplication to find the reflection of (-1,2) about the x axis, y axis and the line y=x?
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### Finding basis of vector spaces

Without proof find the dimension and a basis of the following vector spaces $V$ over the given field $K$. $V$ is the set of all polynomials over $\mathbb{R}$ of degree at most $n$, in which the sum of ...
### For which values of $k$, we have $A = A^{-1}$?
I got this question in hw. Can anyone help me solve it? Let $A = \left( \begin{array}{ccc} k & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & k \end{array} \right)$ For which values of ...