1
to see this. You should get a pivot in each column.
1
(c) No, they do not form a basis because they do not span P3. (You would need 4 vectors to span P3.)
3. Let M2×2 be the vector space of 2×2 matrices with the usual operations of addition and scalar multiplication.
Define the linear transformation
T : M2×2 → M2×2
T(A) = A + A
T
,
where AT
is the transpose of A.
(a) Find the matrix representation of T relative to the basis ��1 0
0 0�
,
�
0 1
0 0�
,
�
0 0
1 0�
,
�
0 0
0 1��.
(b) Find the dimension of the kernel of T.
Solution.
(a) The matrix representation is
2 0 0 0
0 1 1 0
0 1 1 0
0 0 0 2
.
(b) After row reducing the matrix from part (a) there is one free variable. So the dimension of the kernel is
1.
4.
(a) Compute the determinant
det
2 5 −3 −1
3 0 1 −3
−6 0 −4 9
4 10 −4 −1
.
(b) Let A, B be n × n matrices, such that B is invertible. Is it true that
det(B
−1AB) = det(A)?
Justify your answer!
Solution.
(a) The determinant is 120.
(b) It is true, since
det(B
−1AB) = det(B
−1
) det(A) det(B) = det(B
−1
) det(B) det(A) = det(I) det(A) = det(A).
The diagram below is made up of a right triangle and three squares.
Which of the following is represented by this diagram?
A) p2= r2-m2
B)p2=m2-r2
C)r2=p2-m2
D)r2=m2-p2
1 answer
1 answer