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 = […]

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.

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: $$f(n) = \begin{cases}5 & \text{ if n = 0} \\ f(n-1) + 3 + 2n + 2^n \end{cases}$$ […]

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 […]

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 […]

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 […]

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 […]

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? […]

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 […]

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 […]

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