Mark each of the following True or False.
___ a. All vectors in an orthogonal basis have length 1.
___ b. A square matrix is orthogonal if its column vectors are orthogonal.
___ c. If A^T is orthogonal, then A is orthogonal.
___ d. If A is an n*n symmetric orthogonal matrix then A^2=I .
___ e. If A is an n*n symmetric matrix such that A^2=I, then A is orthogonal.
___ f. If A and B are orthogonal n*n matrices, then AB is orthogonal.
___ g. Every orthogonal matrix has nullspace {0}
___ h. Every n*k matrix A has a factorization A=QR, where the column vectors of Q form an orthonormal set and R is an invertible k*k matrix.
To mark each of the statements as True or False, we will go through each statement and make an explanation for the answer.
a. All vectors in an orthogonal basis have length 1.
The statement is False.
Explanation: In an orthogonal basis, the vectors are orthogonal (perpendicular) to each other, but they do not have to have a length of 1. The only requirement for a basis to be orthogonal is that the vectors are mutually perpendicular.
b. A square matrix is orthogonal if its column vectors are orthogonal.
The statement is False.
Explanation: For a square matrix to be orthogonal, both its column vectors and its row vectors must be orthogonal. If only the column vectors are orthogonal, but the row vectors are not, the matrix is not orthogonal.
c. If A^T is orthogonal, then A is orthogonal.
The statement is True.
Explanation: If the transpose of a matrix A, denoted as A^T, is orthogonal, then it implies that the rows (which are now columns after transposing) of A are orthogonal. Since the rows of A are orthogonal, the column vectors of A are orthogonal, making A orthogonal.
d. If A is an n*n symmetric orthogonal matrix, then A^2 = I.
The statement is True.
Explanation: An orthogonal matrix is a square matrix with orthogonal column vectors. If A is symmetric and orthogonal, it means that A^T = A. Therefore, A * A^T = A^2 = I, where I is the identity matrix.
e. If A is an n*n symmetric matrix such that A^2 = I, then A is orthogonal.
The statement is False.
Explanation: Just because a symmetric matrix A satisfies the equation A^2 = I does not mean that the matrix is orthogonal. Orthogonal matrices have the specific property that A^T * A = I.
f. If A and B are orthogonal n*n matrices, then AB is orthogonal.
The statement is True.
Explanation: If A and B are orthogonal matrices, it means that both A^T * A = I and B^T * B = I. In that case, (AB)^T * (AB) = B^T * A^T * AB = B^T * B * A^T * A = I * I = I. Therefore, AB is orthogonal.
g. Every orthogonal matrix has nullspace {0}.
The statement is True.
Explanation: An orthogonal matrix has full rank, meaning its nullspace consists only of the zero vector. This is because the columns of an orthogonal matrix form an orthonormal set, which means they are linearly independent.
h. Every n*k matrix A has a factorization A = QR, where the column vectors of Q form an orthonormal set and R is an invertible k*k matrix.
The statement is False.
Explanation: This statement describes the QR factorization. However, not every n*k matrix has a full QR factorization. The QR factorization exists only if the column vectors of A are linearly independent. If they are not linearly independent, then a reduced QR factorization can be performed where the column vectors of Q still form an orthonormal set, but R is not square or invertible.