Linear Algebra Answers

Solve the system of equations
12 10x1 + x2 + x3 =
12 10 x1 + x2 + x3 =
12 10 x1 + x2 + x3 =
using Gauss-Jordon method with pivoting.

Use the properties of determinants to evaluate the following determinant: (5)
(b + c)
2
a
2
a
2
b
2
(c + a)
2
b
2
c
2
c
2
(a + b

Consider the following system of equations:
x
1 3x
2 x
3 = 3
x
1 + 5x
2 + 3x
3 = 1
x
1 + 7x
2 + 3x
3 = 1
Check whether the system of equations have a solution or not

Let T : R^2 -> R^2 and S: R^2 -> R^2 be linear operators defined by
T ( x(subscript1) , x(subscript2) ) = (x(subscript1) + x(subscript2) , x(subscript1) - x(subscript2)) and S( x(subscript1) , x(subscript2) ) = ( x(subscript1) , x(subscript1) + 2x(subscript2) )
respectively.
i) Find ToS and SoT.
ii) Let B ={ (1;0) , (0;1) } be the standard basis of R^3. Verify that
[ToS](subscriptB) = [T](subscriptB) o [S](subscriptB).

solve the set of linear equation a+2b+3c=5,3a-b+2c=8,4a-6b-4c=2 find c

Let P^3 ={ax^3+bx^2+cx+d ! a,b,c,d ϵ R}. Check whether f (x) = x^2+2x+1 is in[S],
the subspace of P^3 generated by S ={3x^2+1, 2x^2+x+1}.

If f (x) is in [S], write f as a linear combination of elements in S.

If f (x) is not in [S], find anotherpolynomial g(x) of degree at most two such that f (x)
is in the span of S U {g(x)}.

Alsowrite f as a linear combination of elements in S U {g(x)}.

Let V ={(a,b,c,d) ϵ R^4! a+b+2c+2d = 0} and W ={(a,b,c,d) ϵ R4! a = -b;c = -d}

Find the dimensions of V and W.
.

Let V ={(a,b,c,d) ϵ R^4!a+b+2c+2d = 0} and W = { (a;b;c;d) ϵ R^4!a = -b;c = -d}.
Check that V and W are vector spaces.
Further, check that W is a subspace of V.

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

There is no matrix which is Hermitian as well as Unitary.

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

No skew-symmetric matrix is diagonalisable.