Linear Algebra

Homework 10 20 points

Put answers in spaces provided. Please use pencil. Show work.

Any non-zero vector can be the start of defining an orthonormal basis for R2. First we normalize the vector by multiplying by , then multiply the vector by to get the second vector in the basis . Perform these operation on the given vectors. (One point for each vector in the orthonormal basis. All square roots should be simplified and the fractions should not have square roots in the denominators.)

First vector: Normalized first vector: [ _____ _____ ]

Second vector: [ _____ _____ ]

First vector: Normalized first vector: [ _____ _____ ]

Second vector: [ _____ _____ ]

Two linearly independent vectors in R3 define a plane in three dimensional space. To make an orthonormal basis for this plane, project the first vector onto the second vector using the formula , then subtract the projection from the original vector. This gives us two orthogonal vectors, which then need to be normalized. (See the last problems of Homework 5 for the first parts of this process.) To save space, these vectors are written as row vectors instead of column vectors. (8 points)

Let = [3 6 -2] and = [4 -3 6]

= ____________________ = ________________

Normalize these two vectors: ______________________________________

= ____________________ = ________________

Normalize these two vectors: ______________________________________

(continued on back)

Take the two pairs of normalized vectors from the second section of page 1 and perform the cross product to find a third orthonormal vector to complete a basic for all of R3. If [a b c] and [d e f], the cross product . (2 points)

Cross product of the first two orthonormal vectors: ____________________________

Cross product of the second two orthonormal vectors: _________________________

The Gram-Schmidt process taught in class can be used to make an orthonormal basis from any set of linearly independent vectors in Rn. Use the method to make an orthonormal basis using these four vectors in R4, all of which are of unit length. Show work.(6 points)