Linear Algebra Answers

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.

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

If a linear operator is diagonalisable, its minimal polynomial is the same as the
characteristic polynomial.

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

If zero is an eigenvalue of a linear transformation T, then T is not invertible.

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

If the characteristic polynomial of a linear transformation is (x-1)(x-2), its
minimal polynomial is x-1 or x-2.

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

The row-reduced echelon form of an invertible matrix is the identity matrix.

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

The rank of a matrix equals its number of nonzero rows.