Class 24: Question 1 Which of the following set of vectors is not an orthogonal set?

Class 24: Answer 1: (3) Just take the dot product of each pair and you will see that the only pair which does not produce zero is (3,0,0,2) and (0,1,0,1).

Class 24: Question 2 TRUE or FALSE: If two vectors are linearly independent, they must be orthogonal. 1. TRUE. 2. FALSE.

Class 24: Answer 2 (B) FALSE: If two vectors are linearly independent, they must be orthogonal. If two vectors are linearly independent they just have to not be scalar multiples of each other, but for orthogonality they must have a dot product of zero. Counterexample: (1,1,1) and (2,3,4). Clearly linearly independent with dot product of 9, not zero (so not orthogonal)

Class 24: Question 3 TRUE or FALSE: Any orthogonal set of nonzero vectors that spans a vector space must be a basis for that space. 1. TRUE. 2. FALSE.

Class 24: Answer 3 (A) TRUE: Any orthogonal set of nonzero vectors that spans a vector space must be a basis for that space. If a set of vectors spans a space then the only question we need to answer as to whether it is a basis or not is whether the set is linearly independent or not. If a set of vectors is orthogonal, it must be linearly independent, so yes, any orthogonal set of vectors that spans a space must be a basis for that space.

Class 24: Question 4 Which of the following set of vectors is an orthonormal set?

Class 24: Answer 4 (D) An orthonormal set is a set of vectors that are all magnitude one and are orthogonal to each other. All of the vector pairs are orthogonal, but only one of them consists of unit vectors, the last pair Since

Class 24: Question 5 Let A be a square matrix whose column vectors are not zero and mutually orthogonal. Which of the following are true? 1.The dot product of any two different column vectors is zero. 2.The set of column vectors is linearly independent. 3.det(A) is not equal to zero 4.For any b, there is a unique solution to Ax=b. 5.All of the above.

Class 24: Answer 5 (E) “A square matrix whose column vectors are not zero and are mutually orthogonal” is an orthonormal matrix. Theorem 5.8 gives us the result that this matrix will have a determinant that will be plus or minus one (i.e. not zero). Thus The dot product of any two different columns must be zero. Since all the columns are orthogonal, they are linearly independent. Since the determinant is not zero, we know the matrix is invertible, and in fact we know that A T =A -1 thus the linear system must have a unique solution. Therefore all the statements are true.