Use the Gram-Schmidt process starting with the bases B = {1, x, x2} for R2[x] to obtain an orthogonal basis. Then make it an orthonormal basis.
Added by David B.
Step 1
Therefore, the orthogonal basis for R2[x] is {1, 0, x2}. Show more…
Show all steps
Close
Your feedback will help us improve your experience
Madhur L and 82 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
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 $B^{\prime \prime}$. $$ B=\{\langle 5,7\rangle,\langle 1,-2\rangle\} $$
Vectors
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 $B^{\prime \prime}$. $$ B=\{\langle 1,1\rangle,\langle 1,0\rangle\} $$
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD