Question

Consider an orthogonal matrix Q ∈ R ^m×m and a diagonal matrix with non-zero diagonal elements D ∈ R^ m×m and answer the True— False questions below.

(k) Any vector x ∈ R^m can be written as a linear combination of the columns of D

(l) The columns of Q span R^m

(m) The columns of D span R^m

(n) The columns of Q are a basis for R^m

(o) The columns of D are a basis for R^m

Answer 1

(k) Since the diagonal entries in are non-zero, the columns are non-zero multiples of the standard basis vectors. Therefore, they form a basis of . In particular, can be written as a linear combination of columns of . (TRUE)

(l) Since the matrix is orthogonal, it is invertible; therefore, the columns are linearly independent and there are columns, hence they form a basis for . In particular, they span . (TRUE)

(m) Since the diagonal entries in are non-zero, the columns are non-zero multiples of the standard basis vectors. Therefore, they form a basis of . In particular, can be written as a linear combination of columns of . (TRUE)

(n) Since the matrix is orthogonal, it is invertible; therefore, the columns are linearly independent and there are columns, hence they form a basis for . (TRUE)

(o) Since the diagonal entries in are non-zero, the columns are non-zero multiples of the standard basis vectors. Therefore, they form a basis of . In particular, can be written as a linear combination of columns of . (TRUE)