Gram-Schmidt Orthogonalization

Normalize the lengths of these vectors.

• (a)  $\begin{pmatrix}1\\ 2\end{pmatrix}$     (b)  $\begin{pmatrix}-1\\ 3\\ 0\end{pmatrix}$     (c)  $\begin{pmatrix}1\\ -1\end{pmatrix}$

Perform Gram-Schmidt on this basis for $\mathbb{R}^{2}$.

Check that the resulting vectors are orthogonal.

Perform the Gram-Schmidt process on this basis for $\mathbb{R}^{3}$.

Perform Gram-Schmidt on each of these bases for $\mathbb{R}^{2}$.

• (a)  $\langle\begin{pmatrix}1\\ 1\end{pmatrix},\begin{pmatrix}2\\ 1\end{pmatrix}\rangle$     (b)  $\langle\begin{pmatrix}0\\ 1\end{pmatrix},\begin{pmatrix}-1\\ 3\end{pmatrix}\rangle$     (c)  $\langle\begin{pmatrix}0\\ 1\end{pmatrix},\begin{pmatrix}-1\\ 0\end{pmatrix}\rangle$

Then turn those orthogonal bases into orthonormal bases.

Perform the Gram-Schmidt process on each of these bases for $\mathbb{R}^{3}$.

• (a)  $\langle\begin{pmatrix}2\\ 2\\ 2\end{pmatrix},\begin{pmatrix}1\\ 0\\ -1\end{pmatrix},\begin{pmatrix}0\\ 3\\ 1\end{pmatrix}\rangle$     (b)  $\langle\begin{pmatrix}1\\ -1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\begin{pmatrix}2\\ 3\\ 1\end{pmatrix}\rangle$

Then turn those orthogonal bases into orthonormal bases.

Find an orthonormal basis for this subspace of $\mathbb{R}^{3}$: the plane $x-y+z=0$.

Find an orthonormal basis for this subspace of $\mathbb{R}^{4}$.

Show that any linearly independent subset of $\mathbb{R}^{n}$ can be orthogonalized without changing its span.

What happens if we try to apply the Gram-Schmidt process to a finite set that is not a basis?

What happens if we apply the Gram-Schmidt process to a basis that is already orthogonal?

Find a nonzero vector in $\mathbb{R}^{3}$ that is orthogonal to both of these.

One advantage of orthogonal bases is that they simplify finding the representation of a vector with respect to that basis.

• For this vector and this non-orthogonal basis for $\mathbb{R}^{2}$

  first represent the vector with respect to the basis. Then project the vector into the span of each basis vector $[\smash{\vec{\beta}_{1}}]$ and $[\smash{\vec{\beta}_{2}}]$.

• With this orthogonal basis for $\mathbb{R}^{2}$

  represent the same vector $\vec{v}$ with respect to the basis. Then proj…

Bessel’s Inequality. Consider these orthonormal sets

along with the vector $\vec{v}\in\mathbb{R}^{4}$ whose components are $4$, $3$, $2$, and $1$.

• Find the coefficient $c_{1}$ for the projection of $\vec{v}$ into the span of the vector in $B_{1}$. Check that $\lvert\vec{v}\,\rvert^{2}\geq\lvert c_{1}\rvert^{2}$.

• Find the coefficients $c_{1}$ and $c_{2}$ for the projection of $\vec{v}$ into the spans of the two vectors in $B_{2}$. Check that $\lvert\vec{v}\,\rvert^{2}\geq\lvert c_{1}\rvert^{2}+\lvert c_{2}\rvert^{2}$.

• Find $c_{1}$, $c_{2}$, and $c_{3}$ associated with the vectors in $B_{3}$, and $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$ for th…

Prove or disprove: every vector in $\mathbb{R}^{n}$ is in some orthogonal basis.

Show that the columns of an $n\!\times\!n$ matrix form an orthonormal set if and only if the inverse of the matrix is its transpose. Produce such a matrix.

Does the proof of Theorem 2.2 fail to consider the possibility that the set of vectors is empty (i.e., that $k=0$)?

Theorem 2.7 describes a change of basis from any basis $B=\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{k}\rangle$ to one that is orthogonal $K=\langle\vec{\kappa}_{1},\ldots,\vec{\kappa}_{k}\rangle$. Consider the change of basis matrix ${\rm Rep}_{B,K}(\operatorname{id})$.

• Prove that the matrix ${\rm Rep}_{K,B}(\operatorname{id})$ changing bases in the direction opposite to that of the theorem has an upper triangular shape — all of its entries below the main diagonal are zeros.

• Prove that the inverse of an upper triangular matrix is also upper triangular (if the matrix is i…

Complete the induction argument in the proof of Theorem 2.7.