Inner Product Spaces
Introuction to Inner Product SpaceÂ
Last Updated: June 26, 2024Inner Product Space Definition: Let \(\pmb{ V}\) be vector space over a field \(\pmb{ F}\). An \(\fcolorbox{blue}{white}{\pmb{Inner Product}}\) on \(\pmb{ V}\) is a mapping from \(\pmb{ V \times V \to F}\), which is denoted by \( \pmb{\lang x,y\rang~\forall~ x,~y\in V }\), that satisfies the following conditions: a) \( \pmb{\lang x+y,z\rang=\lang x,z\rang+\lang y,z\rang}\) \( \pmb{\forall~...
Properties of Inner Products
Last Updated: June 26, 2024Properties of Inner Products Statement-1: Let \(\pmb{ V}\) be inner product space over a field \(\pmb{ F}\). Then prove that \( \pmb{\lang x,y+z\rang=\lang x,y\rang+\lang x,z\rang}\) \( \pmb{\forall~ x,~y,~z\in V }\) Proof: Let \( \pmb{ x,~y,~z\in V }\), then \begin{align*} \pmb{\lang x,y+z\rang}& = \pmb{\overline{\lang y+z,x\rang}~since~\overline{\lang x,y\rang}=\lang y,x\rang}\\ &= \pmb{\overline{\lang y,x\rang+\lang z,x\rang}~since~\lang x+y,z\rang=\lang x,z\rang+\lang y,z\rang}\\ &=...
Norm of Vector in Inner Product Space
Last Updated: July 1, 2024Norm of Vector Definition: Let \(\pmb{ V}\) be inner product space over a field \(\pmb{ F}\) and \(\pmb{x\in V}\). Then the Norm of the vector \(\pmb{x}\) is denoted by \(\pmb{||x||}\) and is define by \(\pmb{||x||=\sqrt{\lang x,x \rang}}\) Example: Let \(\pmb{ V=F^{n}}\) and \(\pmb{x\in F^{n}}\) such that \(\pmb{x=(x_{1},x_{2},…,x_{n})}\) then the norm of \(\pmb{x}\) is...
Conjugate Transpose of a Matrix
Last Updated: July 1, 2024Conjugate Transpose of a Matrix Definition: Let \(\pmb{A\in M_{m\times n} (F)}\) be a \(\pmb{m\times n}\) matrix. Then the \( \fcolorbox{blue}{white}{\pmb{Conjugate Transpose}}\) or \( \fcolorbox{blue}{white}{\pmb{Adjoint}}\) of \(\pmb{A}\) is denoted by \(\pmb{A^{*}}\) and is defined by the \(\pmb{n\times m}\) matrix such that \(\pmb{A^{*}_{ij}=\overline{A_{ji}} }\) Example-1: Let \(\pmb{A=\begin{bmatrix} 2+3i & -3+2i \\ 7 & 5i \end{bmatrix} }\)...
Orthogonal and Orthonormal Sets
Last Updated: July 1, 2024Orthogonality and Orthonormality Orthogonal Vectors Definition: Let \(\pmb{V}\) be a inner product space over a field \(\pmb{F}\) and let \(\pmb{x,y}\) be two vectors in \(\pmb{V}\). The vectors \(\pmb{x}\) and \( \pmb{y}\) are said to be \(\fcolorbox{blue}{white}{\pmb{orthogonal}}\) if \( \fcolorbox{blue}{white}{\(\pmb{\lang x,y \rang =0}\)}\). Example: Let \(\pmb{x,y\in \mathcal{R}^{2}}\) such that \(\pmb{x=(1,2)}\) and \(\pmb{y=(2,-1)}\) then \(\pmb{\lang...
