Linear Algebra Answers

Determine the polynomial function whose graph passes through the points (2, 4), (3,6) and (5,10). Also sketch the graph of the polynomial function. (Using Cramers Method).

Find an orthogonal matrix P whose first row is u1=(1/3, 2/3, 2/3)

Which of the following statements are true ?
Give reasons for your answers.
(a) If A belongs to Mn(R), then rank (A)=det(A).
(b) There is one and only one unitary matrix in
Mn(R).
(c) If U and V are subspaces of a vector space
W over R, then U U V is also a subspace of W.
(d) Given a linear transformation T from
R4 to R6, rank (T) + nullity (T) = 6.
(e) The relation R, defined on the set of lines in
R² by `L1 R L2 iff L1 and L2 intersect', is an
equivalence relation.

Let V be the vector space of all n x n
matrices over R. What is the dimension of
V over R ? Further, let
Wn={Anxn belongs to V I Anxn is upper triangular}.
Check whether or not W is a subspace of V.

If

A=[ 1 -3 2
4 1 -1
-3 2 5 ]
and

B=[ 2 1 5
-1 -2 -2
3 1 2 ]

then find (AB)^-1

Find the characteristic equation of the matrix

A= [ 1 2 3
4 1 -1
-3 2 5 ]
and verify Cayley-Hamilton theorem for it.

if A= [ 1 -3 2 and B= [ 2 1 5
4 1 -1 -1 -2 -2
-3 2 5 ] 3 1 2 ]

then find (AB)^-1

Solve the system using Gaussian elimination with back-substitution.

3x1 + 2x2 -x3= -15
5x1+ 3x2 + 2x3 =0
3x1+ x2 +3x3=11
11x1+7x2 = -30

Find a basis and the dimension of the subspace w of V spanned by the matrices

A=[ 1 2
-1 3]
B=[ 2 5
1 -1]
C=[3 4
-2 5]

Determine whether or not the following vectors span R^3

u1 = ( 1, 1, 2), u2 = ( 1, -1, 2 ) and u3 = ( 1, 0, 1 ).