Linear Algebra Answers

Solve the following equations using matrix algebra:

2x + y - z = 11

x - 2y + 2z = -2

3x - y + 3z = 5

P(e)(x) = {p(x) ∈ R[x]|p(x) = p(−x)}
Find W = P(e) ∩P3. Find a basis for W and find the dimension of W

Show the cartesian product of R^n and R^m, is isomorphic to R^(n+m)

Let V = R3
, A = {(x, y,z)|y = 0} and B = {(x, y,z)|x = y = z}. Check whether
R3 = A⊕B

a.find a basis for the span of the following set of vectors
|1 | |-4 | |1 | |5| |2| | -1| |11|
|3 | |-12| |-4| |8| |11| | -7| | 52 |
|-1| | 4 | |-5| |-17| |-21| |19| | -80|
|2| |-8| |4| |16| |17| | -3| | 82|

b. find the coordinate vector [x]B for the vector(see below) using the basis from a above
|-8|
|-58|
|151|
|-58|

Solve the following
0.6x + 0.8y + 0.1z = 1
1.1 x + 0.4y + 0.3z= 0.2
x + y + 2z =0.5
by LU decomposition method and find the inverse of the coefficient matrix

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.
i) A =


−2 −5 −1
3 6 1
−2 −3 1

 ii) B =


−1 −3 0
2 4 0
−1 −1 2

.
b) Find inverse of the matrix B in part a) of the question by finding the adjoint as well
as using Cayley-Hamiltion theorem.

 ii) B =


−1 −3 0
2 4 0
−1 −1 2

.
b) Find inverse of the matrix B in part a) of the question by finding the adjoint as well
as using Cayley-Hamiltion

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}.