Linear Algebra Answers

2. Let A="\\begin{pmatrix}\n - 3 & 1 & 0\\\\\n - 6 & 2 & 0\\\\\n - 3 & 1 & 0 \n\\end{pmatrix}"

a) Find the characteristic polynomial of A and show that the eigenvalues are 0 and - 1

b) Find a basis for each eigenspace of A

c) Explain why is A diagonalisable

d) Find an invertible matrix P and a diagonal matrix D such that A=PDP-¹

e) Hence, or otherwise, calculate A²⁰¹⁸

3. In each of the following cases explain whether R²"\\to" R is a linear transformation, if it is, supply a proof, if not, supply a counter ample

a) T(a, b) =a + b

b) T(a, b) =ab

c) T(a, b) =|a|²

d) T(a, b) =a - b

4.a) Let R[x] deg≤2 be the vector space of all polynomial functions in a single variable x with real coefficient and degree at most 2

Let R[x] deg≤2"\\to"R[x] deg≤2 be defined by T(p(x)) =p(x-1)

Let U = <1,x,x²> be the usual basis for R[x] deg≤2 and B = <1+x+x²,2x+x²,x+x²> be another basis for R[x] deg≤2

Let, for any polynomial p(x) "\\isin" R[x] deg≤2, coord_B(p(x)) R³ is the coordinates of p(x) with respect to the basis B for instance

coord_V(x+x²) =(0, 1,1)

and

coord_B(x+x²)=(0, 0,1)

Also recall the agreement every vector X=(x₁, x₂, x₃) "\\isin" R³ is considered as a 3x1 matrix "\\begin{pmatrix}\n X1 \\\\\n X2 \\\\\n X3 \n\\end{pmatrix}" i.e, as a column vector

a) Find the matrix Mat_U"\\to" _U (T) representing the linear transformation T with respect to the basis of U

b) Verify the equation Mat_U"\\to" _U (T)coord_U(p(x)) =coord_U(T(p(x)), for any p(x) "\\isin" R[x] deg≤2

c) Obtain the change of basis matrix P_B"\\to"_U from B to U

d) Hance, or otherwise, obtain coord_B(p(x))for each p(x) "\\isin" R[x] deg≤2

5.a) Given a vector space V, when is a subset T ⊆ V of V said to be a vector subspace of V? 

b) Given a linear transformation V "\\to" W show that the kernel Ker f of f is a subspace of V

c) Given any linear transformation

V "\\to" W show that the subsetI m[f] ={y∈W y=f(x), for some x∈V} is a vector subspace of W

d) Given any linear transformation V "\\to" W, show that f is one-to-one, if and only if, its kernel is trivial i.e, Ker f={0}

e) Given a linear transformation V "\\to" W between finite dimensional vector spaces, show that dim Ker f + dim Im[f] =dim V

Hint: Show Ker f is finite dimensional and then starting with a basis B_o for Ker f extend to a basis of V

6. a) Given any real symmetric square matrix A of order n, show that the function RⁿxRⁿ "\\to" defined by I(x, y) =x^TAy is a symmetric bilinear function

Give examples of matrices A for which the corresponding I is not an inner product

State the extra properties on A which would make the corresponding I an inner product

b) Show that R[x] deg ≤ 2 x R[x] deg ≤ 2 defined by I (p(x), q(x)) =p(0)q(0)+p(1/2)q(1/2)+p(1)q(1) is an inner product on R[x] deg ≤ 2

Use Gram-Schmidt procedure to obtain a orthogonal basis of R[x] deg ≤ 2(with respect to the inner product defined above) which contains the polynomial x

suppose W is plane 3x +6y-4z=0. a basis for W is?

Let n"\\in"N consider the set of nxn symmetric matrices over R with the usual addition and multiplication by a scalar

3.1. Show that this set with the given operations is a vector subspace of M_nn

3.2. What is the dimension of this vector subspace?

3.3. Find a basis for the vector space of 2x2 symmetric matrices.

Consider the vector space P₃

_5.1. Is span {1+x,x+x²,x²+x³,x³+1}=P₃ Motivate your answer

5.2. Let D P₃→ P₂ be the differentiation operator D(a₀+a₁x+a₂x²+a₃x³)=a1+2a₂x+3a₃x²

(i) Find the matrix representation of D relative to the basis {1,x,x²,x³} using the coefficient ordering a0+a₁x+a₂x²+a₃x³→"\\begin{bmatrix}\n a0 \\\\\n a1\\\\\n a2 \\\\\n a3\n\\end{bmatrix}"

(ii) Find the kernel and range of D

Consider the matrix A="\\begin{bmatrix}\n 1 & 0 & 2 \\\\\n 0 & 1 & 0 \\\\\n 2 & 0 & 1\n\\end{bmatrix}"

1.1. Show that the characteristic equation for the eigenvalues "\\lambda" of A is given by (λ²-1) (λ-3)=0.

1.2. Find an orthogonal matrix P which diagonalizes A.

1.3. Find Aⁿ (for n ∈ N) as a matrix

Express M as a linear combination of the matrices A, B, C, where M = [

4 7

7 9

] , A = [

1 1

1 1

] , B

= [

1 2

3 4

] , C = [

1 1

4 5

] .