Solution Found!
be an orthonormal set. Verify the following equality by
Chapter , Problem 2E(choose chapter or problem)
Let \(\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}\) be an orthonormal set. Verify the following equality by induction, beginning with p = 2. If \(\mathbf{x}=c_{1} \mathbf{v}_{1}+\cdots+c_{p} \mathbf{v}_{p}\), then
\(\|\mathbf{x}\|^{2}=\left|c_{1}\right|^{2}+\cdots+\left|c_{p}\right|^{2}\)
Questions & Answers
QUESTION:
Let \(\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}\) be an orthonormal set. Verify the following equality by induction, beginning with p = 2. If \(\mathbf{x}=c_{1} \mathbf{v}_{1}+\cdots+c_{p} \mathbf{v}_{p}\), then
\(\|\mathbf{x}\|^{2}=\left|c_{1}\right|^{2}+\cdots+\left|c_{p}\right|^{2}\)
ANSWER:Solution 2E
Step 1 of 2
The orthonormal set is .
So, the magnitude of each vector is zero and dot product of each pair of vectors is 0.
Let
The objective of solution is to prove
Use principle of induction to prove this result.
If is an orthonormal set then the vectors and are orthogonal.
Because,
Let
Consider the expression,
So the given equality holds for
