Articles of linear algebra

Indefinite inner product spaces?

Hilbert spaces require a positive definite inner product and are very well studied. Have vector spaces with a real “inner product” been studied at all, and if so under what name? In other words $\langle x|y\rangle $ is a real number, not just positive definite, and $\langle x|x\rangle = 0$ does not imply $|x\rangle = […]

How to get the co-ordinates of scaled down polygon

Let say I have a polygon. I need to draw another polygon inside this polygon which is scaled-down. See this image, I need inner Polygon co-ordinates. Given that I have outer polygon co-ordinates and scaled down value.

How to get the characteristic equation from a recurrence relation of this form?

I’ve been getting the characteristic equation from relations of the form $$U_n=3U_{n-1}-U_{n-3}$$ Thanks to this question I made before: How to get the characteristic equation? Now, I recently have been asked to get such equation from this: $$f(n) = \begin{cases}5 & \text{ if n = 0} \\ f(n-1) + 3 + 2n + 2^n \end{cases}$$ […]

Is it true that if ${\bf{x}}^\text{T}{\bf{Ay}}={\bf{x}}^\text{T}\bf{By}$ for all $\bf{x},\bf{y}$ then $\bf{A}=\bf{B}$?

I am puzzled by the following problem: Suppose all vectors and matrices are defined on $\Bbb R ^n$ and $\bf A,B$ are both $n\times n$ square matrices. Is it true that if ${\bf x}^\text{T} {\bf Ay} = \bf{x}^\text{T}\bf By$ for all $\bf x,y$ then $\bf A = B $? I tried to prove this without […]

Product of permutation matrices is the matrix of the composition

I want to prove that the product of two permutation matrices equals the matrix of the composition of the two permutations: $M(\sigma).M(\tau)=M(\sigma \circ \tau)$, where both $M(\sigma)$ and $M(\tau)$ are $n\times n$ So far this is what I got: $M(\sigma) = (s_{ij})$ and $M(\tau)= (t_{ij})$ $M(\sigma).M(\tau)=A$ $A=(a_{ij})$, where: $a_{ij} = \sum_{k=1}^{n}s_{ik}\times t_{kj}$ Now is where […]

Projections onto ranges/subspaces

I’m stuck on a review problem. Consider the matrix: $$\left[ \begin{array}{ccc} -1 & 1 \\ 1 & 1\\ 2 & 1 \end{array} \right] $$ I’m asked to find a matrix $P$ which projects onto the range of $A$, with respect to the standard basis. I’m not actually sure how to tackle this problem. I know […]

Show that $T-iI$ is invertible when $T$ is self-adjoint

Let $T$ be a self adjoint operator on a finite dimensional inner product space $V$. Then $ \| T(x)\pm ix \|^2=\| T(x) \|^2+\| x\|^2$ for all $x \in V$. Deduce that $T-iI$ is inverible. Since $\| T(x)\pm ix \|=0$ iff $\| T(x)\|=0$ and $\| x \|=0$ and it means $N(T)=0$ and it is one-to-one. Since […]

A non-square matrix with orthonormal columns

I know these 2 statements to be true: 1) An $n$ x $n$ matrix U has orthonormal columns iff. $U^TU=I=UU^T$. 2) An $m$ x $n$ matrix U has orthonormal columns iff. $U^TU=I$. But can (2) be generalised to become “An $m$ x $n$ matrix U has orthonormal columns iff. $U^TU=I=UU^T$” ? Why or why not? […]

How to find basis of $\ker T$ and $\mathrm{Im} T$ for the linear map $T$?

Find the basis of $\ker T$ and $\mathrm{Im} T$ for the linear map $T:M^{\mathbb R}_{2 \times 2} \to M^{\mathbb R}_{2 \times 2}$ defined as $T(A)=A-A^t$ for all $A \in M^{\mathbb R}_{2 \times 2}$. Let $A=\begin{pmatrix} a&b\\c&d \end{pmatrix}$. Then: $$ T(A)=\begin{pmatrix} a&b\\c&d \end{pmatrix}-\begin{pmatrix} a&c\\b&d \end{pmatrix}=\begin{pmatrix} 0&b-c\\c-b&0 \end{pmatrix}\stackrel{R_2 \gets-1\cdot R_2}{=}\begin{pmatrix} 0&b-c\\b-c&0 \end{pmatrix} $$ In order to find […]

What does this theorem in linear algebra actually mean?

I’ve just began the study of linear transformations, and I’m still trying to grasp the concepts fully. One theorem in my textbook is as follows: Let $V$ and $W$ be vector spaces over $F$, and suppose that $(v_1, v_2, \ldots, v_n)$ is a basis voor $V$. For $w_1, w_2, \ldots, w_n$ in $W$, there exists […]