In: Advanced Math
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
(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)