<aside> 💡 If the angle between two vectors $u$ and $v$ in $\R^n$ is a right angle(90$^{o}$), then the vectors are called orthogonal vectors. Here $u.v=0$.
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<aside> 💡 Two vectors $u$ and $v$ of an inner product space $V$ are said to be orthogonal, if $\langle u,v \rangle=0$,(i.e.,Inner product of two vectors is zero.)
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The notion of orthogonality depends on the chosen inner product, i.e., the vectors in the example are not orthogonal for the inner product: dot product.
By convention, when two vectors are orthogonal, then the inner product is dot product.
<aside> 💡 An orthogonal set of vectors of an inner product space $V$ is a set of vectors whose elements are mutually orthogonal.
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if $S=\{ v_1,v_2,.....,v_k\} \subseteq V$ , then $S$ is an orthogonal set of vectors if:
$\langle v_i,v_j \rangle=0$ for $i,j\in \{ 1,2,....,k\}$ and $i\neq j$
<aside> 💡 Let $\{v_1,v_2,....,v_k\}$ be an orthogonal set of non-zero vectors in the inner product space $V$. Then $\{v_1,v_2,....,v_k\}$ is a linearly independent set of vectors.
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2nd step on the right side.
<aside> 💡 Let $V$ be an inner product space. A basis consisting of mutually orthogonal vectors is called an orthogonal basis.
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Since an orthogonal set of vectors is already linearly independent, an orthogonal set is a basis precisely when it is a maximal orthogonal set(i.e., there is no orthogonal set strictly containing this one.)
<aside> 💡 If $dim(V)=n$,then: orthogonal basis $\equiv$ orthogonal set of $n$ vectors.
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