To find the matrix M(T) of the linear transformation T: V → V with respect to the basis β = {x₁, x₂}, given the matrix M(T) with respect to the basis β' = {y₁, y₂}, we need to express the basis vectors x₁ and x₂ in terms of y₁ and y₂, and then use the change of basis formula. Given: - V is a two-dimensional vector space over R - T: V → V is a linear map - β = {x₁, x₂} is a basis for V - β' = {y₁, y₂} is another basis for V, where y₁ = x₁ and y₂ = x₁ + x₂ - M(T) with respect to β' is, by rows, [2 -1], [3 1] Step 1: Express x₁ and x₂ in terms of y₁ and y₂. Since y₁ = x₁, we have: x₁ = y₁ Since y₂ = x₁ + x₂, we can rearrange to get: x₂ = y₂ - x₁ = y₂ - y₁ Step 2: Construct the transition matrix P from β' to β. The transition matrix P represents the change of basis from β' to β, and its columns are the coordinate vectors of the basis vectors of β with respect to β'. P = [ [ 1, 0 ], # Coordinate vector of x₁ with respect to β' [ 1, 1 ] # Coordinate vector of x₂ with respect to β' ] Step 3: Find the matrix M(T) with respect to β using the change of basis formula. M(T)_β = P⁻¹ × M(T)_β' × P Where: - M(T)_β is the matrix of T with respect to the basis β - M(T)_β' is the given matrix of T with respect to the basis β' - P is the transition matrix from β' to β - P⁻¹ is the inverse of P First, we need to find P⁻¹: P⁻¹ = [ [ 1, -1 ], [ 0, 1 ] ] Now, we can calculate M(T)_β: M(T)_β = P⁻¹ × M(T)_β' × P = [ [ 1, -1 ], [ 0, 1 ] ] × [ [ 2, -1 ], [ 3, 1 ] ] × [ [ 1, 0 ], [ 1, 1 ] ] = [ [ 2, 1 ], [ 3, 0 ] ] Therefore, the matrix M(T) of the linear transformation T: V → V with respect to the basis β = {x₁, x₂} is: M(T)_β = [ [ 2, 1 ], [ 3, 0 ] ]