Question
Construct the first three orthogonal basis elements for sample points $t_1, \ldots, t_m$ that are in general position.
Step 1
This means that no subset of these points can be perfectly fitted by a polynomial of degree less than the size of the subset minus one. Show more…
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Use the Gram-Schmidt orthogonalization process (3) to transform the given basis $B=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ for $R^{2}$ into an orthogonal basis $B^{\prime}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$. Then form an orthonormal basis $B^{\prime \prime}=\left\{\mathbf{w}_{1}, \mathbf{w}_{2}\right\}$ (a) First construct $B^{\prime \prime}$ using $\mathbf{v}_{1}, \mathbf{u}_{1}$. (b) Then construct $B^{\prime \prime}$ using $\mathbf{v}_{1}, \underline{u}_{2}$. (c) Sketch $B$ and each basis $B^{\prime \prime}$. $$ \boldsymbol{B}=\{\langle-3,4\rangle,\langle-\mathbf{1}, 0\rangle\} $$
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Gram-Schmidt Orthogonalization Process
Use the Gram-Schmidt orthogonalization process (3) to transform the given basis $B=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ for $R^{2}$ into an orthogonal basis $B^{\prime}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$. Then form an orthonormal basis $B^{\prime \prime}=\left\{\mathbf{w}_{1}, \mathbf{w}_{2}\right\}$ (a) First construct $B^{\prime \prime}$ using $\mathbf{v}_{1}, \mathbf{u}_{1}$. (b) Then construct $B^{\prime \prime}$ using $\mathbf{v}_{1}, \underline{u}_{2}$. (c) Sketch $B$ and each basis $B^{\prime \prime}$. $$ B=\{\langle-3,2\rangle,\langle-1,-1\rangle\} $$
Use the Gram-Schmidt orthogonalization process (3) to transform the given basis $B=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ for $R^{2}$ into an orthogonal basis $B^{\prime}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$. Then form an orthonormal basis $B^{\prime \prime}=\left\{\mathbf{w}_{1}, \mathbf{w}_{2}\right\}$ (a) First construct $B^{\prime \prime}$ using $\mathbf{v}_{1}, \mathbf{u}_{1}$. (b) Then construct $B^{\prime \prime}$ using $\mathbf{v}_{1}, \underline{u}_{2}$. (c) Sketch $B$ and each basis $B^{\prime \prime}$. $$ B=\{\langle 5,7\rangle,\langle 1,-2\rangle\} $$
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