Linear Algebra Answers

in the following a set V, a field F, which is either R or C, and operations of addition + and scalar multiplication . , are given for alpha element of F and x element of V, we write their multiplication alpha × x as alpha • x, cheak whether V is a vector space over F, with these operations .

1- Find a real root of the equation 3x +sinx −ex = 0 by the method of false position correct to four decimal places.

2- Find the root of the equation xex = cosx in the interval (0, 1) using Regula-Falsi method correct to four decimal places.

3- Find a real root of the equation x3 −2x −5 = 0 by the method of false position correct to three decimal places.

4- Find the root of the equation tanx + tanhx = 0 which lies in the interval (1.6, 3.0) correct to four significant digits using the method of false position.

5- Using the method of false position, find the root of the equation x6 − x4 − x3 − 1 = 0 up to four decimal places.

6- Find a real root of the equation xlog10x = 1.2 by Regula-Falsi method correct to four decimal places.

Find the general solution of the system
2x + 4y + 6z = 0
4x + 5y + 6z = 3
7x + 8y + 9z = 6:

1.) Find the sequence of elementary matrices whose product is:
A=[1/2 -3
2 3/15] << 2 x 2 matrix


2.) Solve the given SLE using LU-Factorization:
x(sub 1) + x(sub 2) + x(sub 3) = 3
x(sub1) - x(sub 2) + 4x(sub 3) = 4
2x(sub 1) + 3x(sub 2) - 5x(sub 3) = 0

Reduce the quadratic form x1^2 + 2X2x3 to canonical form

Let T(x1, x2, x3) = (x1+ x2, x2+ x3 , x1— x3) be
a linear operator on R³. Find its kernel. Show
that T is not onto. Show that (1, 1, 0) is in the
image of T. Also, find two distinct vectors
u1and u2such that T(u1 ) = (2, 2, 0) = T(u2).

Consider the real vector space Mn(R), of all
n x n matrices with entries from the set of
real numbers with respect to the usual
addition and scalar multiplication of
matrices. Find the smallest subspace of
Mn(R) which contains the identity matrix.
Also show that the set of all symmetric
matrices is a subspace of Mn(R).

Consider the real vector space
A = {(a, b, c, d) I a, b, c, d belongs to R, 2a + 3b = c + d}.
Find dim (A). Also find two distinct subspaces
B1. and B2 of R⁴ such that
A direct sum B1= R⁴=A direct sum B2