1.
linear algebra 16 and 9
In this exercise you will prove Theorem 3 on page 233. Let our assumptions be as in the
statement of this theorem and let the notation be as in Exercise 15a where dim V = dim W = n.
Assume that the nullspace of L = {0}. Explain how it follows from Exercise 15a that the
nullspace of M is {0}? How does it follow that the equation L(X) = Y has at least one solution
for all Y E V? Hint: First explain why M is an n x n matrix. Then apply Theorem 9 on page 146.
Assume that the equation L(X) = Y has at least one solution for all Y V. Prove that then this
equation has only one solution. Hint: First prove that for each B in Rn the system MX = B has at
least one solution. Then apply Theorem 9 on page 146. Let V and W be finite-dimensional vector
spaces with dim V = dim W. Let L : V rightarrow V be a linear transformation. Then the
following statements are equivalent: The nullspace of L is {0}. The equation L(X) = Y has at
least one solution for all Y V. L is invertible. Use the
Solution
16...YOU NEED TO GIVE WHAT IS 15 A TO GIVE PROPER RESPONSE.
9.....LET A=[N X N ] DETERMINANT ....
LET THE 2 TWO EQUAL ROWS BE ....SAY I TH . AND J TH .ROW .
SO WE HAVE ORIGINAL DETERMINANT A AS GIVEN ...LET |A|= D ..........1
NOW LET US EXCHANGE I TH . AND J TH. ROWS TO GET A DETERMINANT A'....
LET |A'| = D' .........................................2
SINCE THE ROWS EXCHANGED ARE IDENTICAL AS GIVEN WE SHALL HAVE A
AND A' AS IDENTICAL ......SO ....
|A|=D=|A'=D' .................D=D'..................................3
BUT AS PER PRPERTY OF ROW EXCHANGE WE SHALL HAVE ......
D ' = - D .............................................4

2.
PUTTING EQN.3 IN EQN.4 WE GET.......
D = -D ..............2D=0................OR ................D=0...
THAT IS ...|A| = 0 .............PROVED