Gram-Schmidt

Section 6.4 Gram-Schmidt

Goals.

We will

review the value of orthonormal bases,
use our understanding of projection to produce orthonormal bases, and
recognize this process as a generalization of the cross product.

Subsection 6.4.1 Review of Orthonormal Bases

We have learned to calculate coordinates from the definition, which requires row reducing, and using an inner product. Because this can be much easier, we work to develop a method for producing orthonormal bases.

Figure 6.4.1. Comparison of Non-Orthogonal and Orthogonal Bases

Activity 6.4.1.

In an orthogonal bases each vector is orthogonal to all the other bases. This means that the projection of any basis vector onto another is the zero vector. What does this mean when we project another vector onto an orthogonal basis?

In both sides of Figure 6.4.1 the blue vectors are the basis. The gray (thick) vectors are the projection of the green vector onto each of the basis vectors.

(a)

In the left image the basis vectors (bottom two) are not orthogonal. Are the projections (gray vectors) orthogonal?

(b)

If we add the two projections (gray vectors) is the result the original vector (green)?

(c)

In the right image the basis vectors (on left and bottom) are orthogonal. Are the projections (gray vectors) orthogonal?

(d)

If we add the two projections (gray vectors) is the result the original vector (green)?

Subsection 6.4.2 Procedure

If we start with a non-orthogonal basis that is convenient for some reason, we want a procedure to turn it into an orthogonal basis.

Activity 6.4.2.

In this activity we use projection to clean up basis vectors.

\begin{equation*} W=\{ \vec{w}_1=[1,2], \vec{w}_2=[2,1] \}. \end{equation*}

(a)

Check if \(\vec{w}_1\) and \(\vec{w}_2\) are orthogonal.

(b)

Draw both vectors.

(c)

Let \(\vec{v}=\proj_{\vec{w}_1} \vec{w}_2\text{.}\) Calculate and draw this projection.

(d)

What direction does the (any) projection have?

(e)

Calculate \(\hat{w}_2=\vec{w}_2-\vec{v}\text{.}\)

(f)

Draw this new vector \(\hat{w}_2\text{.}\)

(g)

Check if \(\vec{w}_1\) and \(\hat{w}_2\) are orthogonal.

Activity 6.4.3.

In the previous activity we discovered how to modify one vector to make it orthogonal to one other vector. In this activity we extend that procedure to three or more vectors.

\begin{equation*} W=\{ [1,1,0],[1,0,1],[0,1,1] \}. \end{equation*}

(a)

Use the procedure from the previous activity to construct \(\hat{w}_2\) that is orthogonal to \(\vec{w}_1\text{.}\)

(b)

Use the procedure again to create a vector \(\vec{v}\) from \(\vec{w}_3\) that is orthogonal to \(\vec{w}_1\text{.}\)

(c)

Use the procedure again to create a vector \(\hat{w}_3\) from \(\vec{v}\) that is orthogonal to \(\hat{w}_2\text{.}\)

(d)

Check if \(\hat{w}_3\) is still orthogonal to \(\vec{w}_1\text{.}\)

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