Reference no: EM132458126

Assignment - Linear Algebra Questions

Q1. (a) Complete the definition: Vectors (v₁)^→, . . . , (v_k)^→ are linearly independent if...

(b) Suppose A is an m × n matrix whose columns are linearly independent. What is the nullity of A?

Q2. Let A denote the matrix

Let T: R² → R² be the linear transformation given by T(x) = Ax.

(a) Describe T geometrically.

(b) Find the characteristic polynomial of A, and use it to find all eigen-values of A or to show that none exist. Explain why your answer makes sense geometrically.

(c) Compute A²⁰¹¹. (Hint: What power of A is equal to the identity?)

Q3. (a) Compute the inverse of the matrix

(b) Find all solutions to the system of linear equations

-4x + 5z = -2

-3x - 3y + 5z = 3

-x + 2y + 2z = -1

Q4. Using Gaussian elimination, find all solutions to the following system of linear equations:

2x₂ + 3x₃ + 4x₄ = 1

x₁ - 3x₂ + 4x₃ + 5x₄ = 2

-3x₁ + 10x₂ - 6x₃ - 7x₄ = -4

Q5. Let A denote the matrix

(a) Find the eigen-values of A.

(b) Find an orthonormal basis of R³ consisting of eigenvectors for A.

(c) Find a 3 × 3 orthogonal matrix S and a 3 × 3 diagonal matrix D such that A = SDS^-1.