Linear Algebra Answers

Consider C^n, with the standard inner product. Find an orthonormal basis for the subspace spanned by v1=(1, 0, i) and v2=(2, 1, 1+i)

Determine whether or not the following vectors span
u1 = ( 1, 1, 2), u2 = ( 1, -1, 2 ) and u3 = ( 1, 0, 1 ).

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