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- Jan 26, 2012

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Thanks to those who participated in last week's POTW!! Here's this week's problem!

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In this problem, the inner product on $P_2$ is given by

\[\langle f,g\rangle = \int_0^1 f(t)g(t)\,dt.\]

Recall that if we have a basis $\{\mathbf{u}_1,\ldots,\mathbf{u}_n\}$, then the orthogonal basis obtained by applying Gram-Schmidt consists of vectors $\{\mathbf{v}_1,\ldots, \mathbf{v}_n\}$ where

\[\mathbf{v}_1=\mathbf{u}_1\qquad\text{and}\qquad \mathbf{v}_k = \mathbf{u}_k-\sum_{i=1}^{k-1}\frac{\langle \mathbf{u}_k,\mathbf{v}_i\rangle}{\langle\mathbf{v}_i,\mathbf{v}_i\rangle}\mathbf{v}_i;\qquad k=2,\ldots, n\]

The orthonormal basis would then be $\{\mathbf{w}_1,\ldots,\mathbf{w}_n\}$ with $\mathbf{w}_j= \dfrac{\mathbf{v}_j}{\langle \mathbf{v}_j,\mathbf{v}_j\rangle}$.

Remember to read the POTW submission guidelines to find out how to submit your answers!

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**Problem**: Apply the Gram-Schmidt process to the basis $\{1,t,t^2\}$ for the Euclidean space $P_2$ (the space of degree 2 or less polynomials with real coefficients) and obtain an orthonormal basis for $P_2$.-----

In this problem, the inner product on $P_2$ is given by

\[\langle f,g\rangle = \int_0^1 f(t)g(t)\,dt.\]

**Remark**:\[\mathbf{v}_1=\mathbf{u}_1\qquad\text{and}\qquad \mathbf{v}_k = \mathbf{u}_k-\sum_{i=1}^{k-1}\frac{\langle \mathbf{u}_k,\mathbf{v}_i\rangle}{\langle\mathbf{v}_i,\mathbf{v}_i\rangle}\mathbf{v}_i;\qquad k=2,\ldots, n\]

The orthonormal basis would then be $\{\mathbf{w}_1,\ldots,\mathbf{w}_n\}$ with $\mathbf{w}_j= \dfrac{\mathbf{v}_j}{\langle \mathbf{v}_j,\mathbf{v}_j\rangle}$.

Remember to read the POTW submission guidelines to find out how to submit your answers!

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