Lecture 28 ORTHOGONAL COMPLEMENTS AND ORTHOGONAL MATRICES Class

Class participation question from 04/07 What happens if one runs the Gram-Schmidt algorithm for

Orthogonal Complements (that’s how it looks like in mathematical symbols) An important observation, which

Orthogonal Complements: the general approach Example. Compute the orthogonal complement of Solution. Solve a

Something to recall from Calc III: (you can easily verify this via linear algebra)

Orthogonal Complements continued Answer. Some additional properties: the intersection

The fundamental spaces of a matrix Read it row-by-row; say, (to derive this, just

Orthogonal matrices Definition. An orthogonal matrix is a matrix of a rotation. rotate Why?

Orthogonal matrices continued E. g. , Before we move on, notice that (dot product)

Orthogonal matrices continued Example. Verify whether a matrix is orthogonal: (i. e. , that

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Lecture 28 ORTHOGONAL COMPLEMENTS AND ORTHOGONAL MATRICES

Class participation question from 04/07 What happens if one runs the Gram-Schmidt algorithm for a basis that’s already orthogonal? Will the Gram-Schmidt algorithm output the same vectors in this case (yes or no)? (think about the first question, submit yes or no for the second one) Answer. Yes, the algorithm outputs the same vectors.

Orthogonal Complements (that’s how it looks like in mathematical symbols) An important observation, which is true in general:

Orthogonal Complements: the general approach Example. Compute the orthogonal complement of Solution. Solve a linear system: (solve as usual: find RREF, choose parameters, etc) Answer:

Something to recall from Calc III: (you can easily verify this via linear algebra) Coefficients form a normal vector to this plane (a line spanned by the normal vector) This works in both ways: if you start with a line, then any non-zero vector on that line determines the plane that’s orthogonal to the line)

Orthogonal Complements continued Answer. Some additional properties: the intersection

The fundamental spaces of a matrix Read it row-by-row; say, (to derive this, just take the transpose)

Orthogonal matrices Definition. An orthogonal matrix is a matrix of a rotation. rotate Why? Recall the meaning of the columns: But what happens when we rotate the standard basis? all pieces stay intact while the vectors are being rotated We get again an orthonormal set: the lengths are preserved by 1, the angles are preserved by 2.

Orthogonal matrices continued E. g. , Before we move on, notice that (dot product) dot product matrix product (matrix product) An important remark: any of the four properties is a defining property: if a matrix satisfies one of those, it’s orthogonal and satisfies all the other properties.

Orthogonal matrices continued Example. Verify whether a matrix is orthogonal: (i. e. , that this matrix is a matrix of a rotation) Solution. The quickest way is to check whether the columns form an orthonormal basis. Example. Verify whether a matrix is orthogonal: Solution. The columns are orthonormal