MATH 2213

FINAL EXAMINATION December 1996 | TIME: 3 HOURS |

**SHOW ALL WORK. YOUR METHOD IS MORE IMPORTANT THAN THE RIGHT
ANSWER.**

- Let
(a) Find

*A*^{-1}using row reduction.(b) Find elementary matrices

*E*_{1},*E*_{2}, . . . ,*E*(_{k}*k*is however many elementary matrices that you need) such that*E*_{k}. . . E_{2}*E*_{1}*A*=*I*.(c) Check your result by calculating

*E*_{k}. . . E_{1}. - Let , and
*B*= {**b**_{1},**b**_{2},**b**_{3}}.(a) Prove that

*B*is a basis for . Quote any results you might use.(b) Find [

**y**]_{B}.(c) Find a matrix

*P*such that*P*[**x**]_{B}=**x**and a matrix*Q*such that [**x**]_{B}=*Q*[**x**] for all**x**in .

(You may use part (c) to do part (b) if you wish.) - Let
(a) Find a basis for Nul

*A*.(b) Find a basis for Col

*A*.(c) Find a basis for Row

*A*.(d) What is the rank of

*A*? - Find a basis for the subspace of spanned by
What is the dimension of this subspace?

- For each of the following, determine the matrix
*A*such that*T***x**=*A***x**.(a) rotates vectors anticlockwise about the origin through an angle of radians.

(b) leaves the line

*y = x*fixed and*T*(*x*, 0) = (2*x*, 0) for all*x*.(c) and

*T*(*x*_{1},*x*_{2},*x*_{3}) = (3*x*_{2}-*x*_{3},*x*_{1}+ 4*x*_{2}+*x*_{3}). - Let
(a) Show that is an eigenvalue and find the corresponding eigenspace.

(b) Show that is an eigenvector for

*A*and find the corresponding eigenvalue. - Let . It can be shown that the eigenvalues of
*A*are 8 and -1 with corresponding eigenspaces(a) What result allows you to immediately conclude that

*A*is diagonalizable?(b) Give a basis

*B*for such that if*T***x**=*A***x**, then [*T***x**]_{B}=*D*[**x**]_{B}where*D*is a diagonal matrix.(c) Produce a matrix

*P*such that*P*^{-1}*AP*is a diagonal matrix. - Which of the following sets are vector spaces? Give reasons
whether the set is or is not a vector space. If it is a vector space,
state the dimension.
(a) All polynomials of the form

*p*(*t*) = 1 +*at*+*bt*^{2}(*a,b*can be any number).(b) All polynomials

*p*(*t*) in*P*_{5}whose derivative is 0 at*t*= 0.

(c) All points in the plane inside the unit circle (i.e. all points such that ). -
(a) If the null-space of a 5 × 6 matrix is 3
dimensional, what are the dimensions of the row and column spaces?
(b) Let

*A*be a 2 × 3 matrix. What are the possible dimensions of the null-space of*A*? Give an example of a 2 × 3 matrix*A*that has each of these possible dimensions. -
(a) Prove that if is given by
, then
*T*is a linear transformation.(b) If

*B*= {1,*t, t*^{2},*t*^{3}} find a matrix*A*such that [*T*(**p**(*t*))]_{B}=*A*[**p**(*t*)]_{B}. That is, find [*T*]_{B}. - Suppose
*S*= {*v*_{1}, . . . ,*v*_{k+1}} is a set of vectors in the vector space**V**, and suppose that*v*_{k+1}is a linear combination of*S'*= {*v*_{1}, . . . ,*v*_{k}}. Prove span{*v*_{1}, . . . ,*v*_{k}} = span{*v*_{1}, . . . ,*v*_{k+1}}.