Find the minimal polynomial of the matrix

Solve the folowing set of simultaneous equations using Cramer's rule:

x+2y+ z = 3

2x - y+2z = 1

3x+y+z=0

Find Adj(A) where A =  .  Hence  find A^-1.

Find the eigenvalues and eigenvectors of the matrix B =  . Is the matrix diagonalisable? Justify your answer.

Show that W = {(x, 4x, 3x) ∈ R²x ∈ R} is a subspace of R³. Also find a basis for subspace U of R³ which satisfies W ⊕ U = R³.

Let T: R³ → R³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

 is  Find the matrix of the linear transformation with respect to the basis

Check that B = {1, 2x + 1, (x - 1)²} is a basis for P_₂, the vector space of polynomials with real coefficients of degree ≤ 2.

Let S be any non-empty set and let V(S) be the set of all real valued functions on R. Define addition on V(s) by (f+g)(x) = f(x) + g(x) and scalar multiplication by (af)(x) = af (x). Check that (V(S), +, -) is a vector space.

Find the signature of the quadratic form 

Find adjoint of the linear operator T: C² → C² defined by T (z₁, z₂) = (z₂, z₁ + iz₂) with respect to the standard inner product on C².

Let C[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner product 

Find the inner product of the functions f(t) = 2t,g(t) = 1/ t²+5. 

Check whether    is an eigenvector for the matrix   What is the corresponding eigenvalue? 

Verify Cayley-Hamilton theorem for the matrix 

Let A be a 2 x 3 matrix, B be a 3 x 4 matrix and C be a 3 x 2 matrix and D be a 3 x 4 matrix. Is AB + C^tD defined? Justify your answer.

Find the matrix of the linear transformation T: R² → R² defined by T(x, y) = (2y, x - y) with respect to the ordered basis {(0, -1), (-1,0)}.

Describe the linear transformation T:R² → R² such that

where B is the standard basis of R².

Find the kernel of the linear transformation T: R² → R² defined by T(x, y) = (2x + 3y, 2x – 3y).

If {U¹, U²} is an ordered basis of R2 and {f_₁ (v), f₂ (v)} is the corresponding dual basis find f₁ (2v₁+v₂) and f₂ (U₁-2v₂).

Check whether T: R² → R², defined by T(x, y) = (-y, x) is a linear transformation.

Check whether the set of vectors {1 + x, x + x², 1 + x³} is a linearly independent set of vectors in P3, the vector space of polynomials of degree ≤ 3.