Question

Let

v₁ = [0 3 -2], v₂ = [-3 -3 0], and v₃ = [2 0 -3]

be eigenvectors of the matrix A which correspond to the eigenavalues λ₁ = -2, λ₂ = 1, and λ₃ = 2, respectively, and let

v - [-2 -3 -8]

Express v as a linear combination of v₁, v₂ and v₃ and find Aᵥ.

v = _____ v₁ + _______ v₂ + ________ v₃

Aᵥ = ________

Answer 1

To express v as a linear combination of v₁, v₂, and v₃, we need to find the coefficients a, b, and c. By solving the system of linear equations, we find that v can be expressed as -3*v₁ + 1*v₂ - v₃. Aᵥ.v, the product of matrix A and vector v, is equal to [-4 -4 -4].

Step-by-step explanation:

To express v as a linear combination of v₁, v₂, and v₃, we need to find the coefficients that multiply each eigenvector to get the sum of the vectors equal to v. Let's call these coefficients a, b, and c:

v = a*v₁ + b*v₂ + c*v₃

Substituting the given values:

[-2 -3 -8] = a*[0 3 -2] + b*[-3 -3 0] + c*[2 0 -3]

To find the values of a, b, and c, we can solve this system of linear equations:

-2 = 0a - 3b + 2c

-3 = 3a - 3b

-8 = -2a + 0b - 3c

Solving this system of equations, we get:

a = -3, b = 1, c = -1

So, v can be expressed as:

v = -3*v₁ + 1*v₂ - v₃

Now, let's find Aᵥ.v:

Aᵥ.v = A*(-3*v₁ + 1*v₂ - v₃)

Expanding this expression using matrix multiplication, we get:

Aᵥ.v = [-2 -1 -3] + [0 -3 2] - [2 0 -3]

Simplifying, we get:

Aᵥ.v = [-4 -4 -4]