## Determinants II: Determinants of Order $n$

A determinant of order $n$ can be calculated by expanding it in terms of determinants of order $n-1$. Let $A=(a_{ij})$ be an $n\times n$ matrix and let us denote by $A_{ij}$ the $(n-1)\times (n-1)$ matrix obtained by deleting the $i$-th … Continue reading

## Determinants I: Determinants of Order 2

Let $A=\begin{pmatrix} a & b\\ c & d \end{pmatrix}$. Then we define the determinant $\det A$ by $$\det A=ad-bc.$$ $\det A$ is also denoted by $|A|$ or $\left|\begin{array}{ccc} a & b\\ c & d \end{array}\right|$. In terms of the column … Continue reading

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## Orthogonal Bases

Let $V$ be a vector space with a positive definite scalar product $\langle\ ,\ \rangle$. A basis $\{v_1,\cdots,v_n\}$ of $V$ is said to be orthogonal if $\langle v_i,v_j\rangle=0$ if $i\ne j$. In addition, if $||v_i||=1$ for all $i=1,\cdots,n$, then the … Continue reading

Let $V$ be vector space. A scalar product is a map $\langle\ ,\ \rangle: V\times V\longrightarrow\mathbb{R}$ such that SP 1. $\langle v,w\rangle=\langle w,v\rangle$ SP 2. $\langle u,v+w\rangle=\langle u,v\rangle+\langle u,w\rangle$ SP 3. If $x$ is a number, then $$\langle xu,v\rangle=x\langle u,v\rangle=\langle … Continue reading Posted in Linear Algebra | Leave a comment ## Sage: Basic Matrix Operations For using Sage: Sage is an open source math software whose interface is a web browser (in particular firefox). You don’t have to install Sage in your computer to use it. You can access any sage server including the main … Continue reading Posted in Linear Algebra, Sage | Leave a comment ## Inverses Let F: V\longrightarrow W be a mapping. F is said to be invertible if there exists a map G: W\longrightarrow V such that$$G\circ F:I_V,\ F\circ G=I_W,$$where I_V: V\longrightarrow V and I_W: W\longrightarrow W the identity maps on V … Continue reading Posted in Linear Algebra | Leave a comment ## Composition of Linear Maps Let F: U\longrightarrow V and G: V\longrightarrow W be two maps. The composite map G\circ F: U\longrightarrow W is defined by$$G\circ F(v)=G(F(v))$$for each v\in U. Example. Let A be an m\times n matrix and let B be a … Continue reading Posted in Linear Algebra | Leave a comment ## The Matrix Associated with a Linear Map Given an m\times n matrix A, there is an associated linear map L_A: \mathbb{R}^n\longrightarrow\mathbb{R}^m as seen here. Conversely, given a linear map L: \mathbb{R}^n\longrightarrow\mathbb{R}^m there exists an m\times n matrix A such that L=L_A. To see this, consider the unit … Continue reading Posted in Linear Algebra | 1 Comment ## The Kernel and Image of a Linear Map Let F:V\longrightarrow W be a linear map. The image of F is the set$$\mathrm{Im}F=\{w\in W: F(v)=w\ \mbox{for some}\ v\in V\}. Proposition. The image of $F$ is a subspace of $W$. Proof. The proof is straightforward. It is left for … Continue reading
Let $V, W$ be two vector spaces. A map $L:V\longrightarrow W$ is called a linear map if it satisfies the following properties: for any elements $u,v\in V$ and any scalar $c$, LM 1. $L(u+v)=L(u)+L(v)$. LM 2. $L(cu)=cL(u)$. That is, linear … Continue reading