Linear Algebra Answers

Let (x1;x2;x3) and (y1;y2;y3) represent the coordinates with respect to the bases
B1 = f(1;0;0); (0;1;0); (0;0;1)g, B2 = f(1;0;0); (0;1;2); (0;2;1)g. If
Q(X) = x2
1+2x1x2+2x2x3+x2
2+x2
3, find the representation of Q in terms of
(y1;y2;y3).

Find the orthogonal canonical reduction of the quadratic form
x2+y2+z2

Let P3 be the inner product space of polynomials of degree at most 3 over R with
respect to the inner product
hf,gi =
Z 1
0
f(x)g(x)dx.
Apply the Gram-Schmidt orthogonalisation process to find an orthonormal basis for
the subspace of P3 generated by the vectors (8)

1−2x,2x+6x
2
,−3x
2 +4x
3

.

Find the orthogonal canonical reduction of the quadratic form
x
2 +y
2 +z
2 −2xy−2xz−2yz. Also, find its principal axes.

Check whether the matrices A and B are diagonalisable. Diagonalise those matrices
which are diagonalisable. (11)
i) A =


−2 −5 −1
3 6 1
−2 −3 1

 ii) B =


−1 −3 0
2 4 0
−1 −1 2

.

Consider the basis e1 = (−2,4,−1), e2 = (−1,3,−1) and e3 = (1,−2,1) of R
3
over R. Find the dual basis of {e1, e2, e3}.

We know that the set F(R) of functions f : R ! R, together with pointwise
addition and scalar multiplication
(f + g)(x) = f(x) + g(x) for all f; g 2 F(R) and x 2 R
(
f)(x) = f(x) for all f 2 F(R),  2 R and x 2 R:
In this problem, you are asked to discuss whether F(R) continues to be a vector space
when the operations (+;
) are replaced by other addition/scalar multiplication opera-

For n > 2 , the AM of the first n natural numbers is greater than n +1.
Is the statement true or false?
Give justification in support of your answer.

X>0 is necessary for x+2>1
Is the statement true or false?
Give justification in support of your answer.

A firm uses two inputs, K and L to manufacture final product. The price per unit of these inputs are sh. 20 and sh. 4 respectively. What combination of inputs should a firm use to maximize output given that the budget is fixed at sh. 390?