Question
Use the inner product $\langle\mathbf{u}, \mathbf{v}\rangle= 2 u_{1} v_{1}+u_{2} v_{2}$ in $R^{2}$ and the Gram-Schmidt orthonormalization process to transform $\{(2,-1),(-2,10)\}$ into an orthonormal basis.
Step 1
Step 1: First, we define the given vectors as $\mathbf{v}_1 = (2,-1)$ and $\mathbf{v}_2 = (-2,10)$. Show more…
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Use the inner product u, v = 2u1v1 + u2v2 in R2 and the Gram-Schmidt orthonormalization process to transform {(2, -1), (2, 10)} into an orthonormal basis. (Use the vectors in the order in which they are given.) U1 = U2 =
Let $R^{3}$ have the inner product $$ \langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+2 u_{2} v_{2}+3 u_{3} v_{3} $$ Use the Gram-Schmidt process to transform $\mathbf{u}_{1}=(1,1,1)$ $\mathbf{u}_{2}=(1,1,0), \mathbf{u}_{3}=(1,0,0)$ into an orthonormal basis.
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Use the Gram-Schmidt orthogonalization process (4) to transform the given basis $\boldsymbol{B}=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ for $R^{3}$ into an orthogonal basis $B^{\prime}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$. Then form an orthonormal basis $B^{\prime \prime}=\left\{\mathbf{w}_{1}, \mathbf{w}_{2}, \mathbf{w}_{3}\right\}$ $$ B=\left\{\left\langle\frac{1}{2}, \frac{1}{2}, 1\right\rangle,\left\langle-1,1,-\frac{1}{2}\right\rangle,\left\langle-1, \frac{1}{2}, 1\right\rangle\right\} $$
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