Linear Algebra. Sample Final 1.
1) Giv en fh : P3 (x) ! M 2;2 de ned b y
fh (a + bx + cx 2 + dx 3 ) =a + b + c + hd b + c
b c hd hb
where h 2 IR is a parameter.
a) Giv e the de n ition of Kernel and Image of a linear transformation. Find, for all p ossible
values of h, Ker( fh ), Im( fh ), their bases and dimensions.
b) Do es there exist a value of h 2 IR suc h th at fh is not an isomorphism?
c) Determine f 1h (A ) (i.e. the set of all p olynomials mapp ed to A ), where A =0 33 0.
d) For h = 0, nd all eigen values and corresp onding eigen vectors of f0 determine if f0 is
diagonalizable.
2) Find the equation of the line p passing through A and orthogonal to the plane con taining the
p oin ts A, B and C . Dete rm ine if the lines p and q are parallel, in tersecting or skew.
A = [1 ; 1; 0], B = [1 ; 2; 2; ], C = [2 ; 0; 4], q : X = [2 ; 4; 1] + t(3 ; 2; 2).
Linear Algebra. Sample Final 2.
1) Giv en the transformation fh : IR3 ! IR3 de ned b y
fh (x; y;z) = (x hz ;x + y hz ; hx + z), where h 2 IR is a parameter.
a) Find, for all p ossible values of h, Ker( fh ), Im( fh ), their bases and dimensions.
b) Is fh an isomorphism for some value of h?
c) Determine f 1h (1 ; 0; 1) = f(x; y;z) 2 IR3 : fh (x; y;z) = (1 ; 0; 1)g.
d) Giv e the de nition of eigen value and eigen vector of a linear transformation. Dete rm ine if fh
is diagonalizable for some values of h. For h = 1, nd a basis of IR3 made of eigen vectors of f1 .
2) Find the pro jection P0 of the p oin t P on the li ne p, the distance of P from p and the co ordinates
of the p oin t R symmetric to P with resp ect to p, where
P = [1 ; 2; 0], and p : X = [3 ; 0; 0] + t(0 ; 1; 0) ;t2 IR .
Linear Algebra. Sample Final 3.
a) Giv e the de nition of regular matrix. Find all values of h for whic h the matrix is regular.
b) For h = 1, solv e the matrix equation A X = B ,
c) Find the solutions of th e system A
A for all p ossible h 2 IR .
d) Is the matrix A diagonalizable for h = 0?
e) For h = 3, nd the Image and the Kern el of the linear transformation l that has A as
asso ciated matrix (with resp ect to the standard basis of IR3 ).
2) Find the distance of the p oin t P from the plane , the pro jection P0 of P on and the volume
of the tetrahedron formed b y P , P0, A and B , where
P = [ 1; 1; 2], : X = [1 ; 1; 1] + t( 3; 2; 2) + u(3 ; 4; 3) ;t; u 2 IR ,
A = [4 ; 5; 4], B = [ 2; 1; 3]