所屬科目：研究所、轉學考(插大)◆線性代數
2. pointsFind a matrix A such thatYou need to show that the matrix you find has the required properties.
3. Let A . Show that the system of linear equations Ax = b has a solution for all b if and only if the system of linear equations has no nonzero solutions.
4. Let A and B . Show that if rank(AB) = m, then rank(BA) =m.
6. Let A . Show that if Q is an invertible matrix such that AQ is diagonal, then each column vector of Q is an eigenvector of .
7. (12 points) Let T: be a linear operator. Show that if T preserves the Euclidqdean distance between any two points, that is, ||T(u) - T(ข)||= ||u- ull for any u, u , then the matrix representation of T relative to the standard basis is an orthogonal matrix.
8. Let A be a real symmetric matrix. Show that there exists al real symmetric matrix B such that B3 = A.
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such that
. Show that the system of linear equations Ax = b has a solution for all b 
if and only if the system of linear equations
has no nonzero solutions.
and B 
. Show that if rank(AB) = m, then rank(BA) =m.
. Show that if Q 
is an invertible matrix such that 
AQ is diagonal, then each column vector of Q is an eigenvector of 
.
be a linear operator. Show that if T preserves the Euclidqdean distance between any two points, that is, ||T(u) - T(ข)||= ||u- ull for any u, u 
, then the matrix representation of T relative to the standard basis is an orthogonal matrix.
be a real symmetric matrix. Show that there exists al real symmetric matrix B such that B3 = A.