Matrix Representation of Linear Transformations - Matrix Algebra, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Matrix transformations

Theorem Suppose L : Rⁿ → R^m is a linear map. Then there exists an m×n matrix A such that L(x) = Ax for all x ∈ Rⁿ . Columns of A are vectors L(e₁), L(e₂), . . . , L(e_n), where e₁ , e₂ , . . . , e_n is the standard basis for Rⁿ .

Basis and coordinates

If {v₁, v₂, . . . , v_n} is a basis for a vector space V , then any vector v ∈ V has a unique representation v = x₁v₁ + x₂v₂ + · · · + x_n v_n , where x_i ∈ R. The coefficients x₁, x₂, . . . , x_n are called the coordinates of v with respect to the ordered basis v₁, v₂, . . . , v_n.

The mapping vector v → its coordinates (x₁, x₂, . . . , x_n) provides a one-to-one correspondence between V and Rⁿ . Besides, this mapping is linear.

Change of coordinates

Let V be a vector space.

Let v₁, v₂, . . . , v_n be a basis for V and g₁ : V → Rⁿ be the coordinate mapping corresponding to this basis.

Let u₁, u₂, . . . , u_n be another basis for V and g₂ : V → Rⁿ be the coordinate mapping corresponding to this basis.

The composition g₂◦g₁⁻¹ is a linear mapping of Rn to itself. It is represented as x → U x, where U is an n×n matrix.

U is called the transition matrix from v₁ , v₂ . . . , v_n to u₁ , u₂ . . . , u_n . Columns of U are coordinates of the vectors v₁ , v₂ , . . . , v_n with respect to the basis u₁ , u₂ , . . . , u_n.

Matrix of a linear transformation

Let V , W be vector spaces and f : V → W be a linear map.

Let v₁, v₂, . . . , v_n be a basis for V and g₁ : V → Rⁿ be the coordinate mapping corresponding to this basis.

Let w₁, w₂, . . . , w_m be a basis for W and g₂ : W → R^m be the coordinate mapping corresponding to this basis.

The composition g₂◦f ◦g₁⁻¹ is a linear mapping of Rⁿ to R^m. It is represented as x → Ax, where A is an m×n matrix.

A is called the matrix of f with respect to bases v₁ , . . . , v_n and w₁, . . . , w_m . Columns of A are coordinates of vectors f (v₁), . . . , f (v_n) with respect to the basis w₁ , . . . , w_m.

Examples. • D : P₃ → P₂, (Dp)(x) = p′(x).

Let A_D be the matrix of D with respect to the bases 1, x , x² and 1, x . Columns of A_D are coordinates of polynomials D1, Dx , Dx² w.r.t. the basis 1, x.

• L : P₃ → P₃, (Lp)(x ) = p(x + 1).

Let A_L be the matrix of L w.r.t. the basis 1, x , x ². L1 = 1, Lx = 1 + x, Lx² = (x + 1)² = 1 + 2x + x².

Problem. Consider a linear operator L on the vector space of 2×2 matrices given by

Find the matrix of L with respect to the basis

Let ML denote the desired matrix.

By definition, M_L is a 4×4 matrix whose columns are coordinates of the matrices L(E₁), L(E₂), L(E₃), L(E₄) with respect to the basis E₁ , E₂ , E₃ , E₄ .

It follows that

Thus the relation

is equivalent to the relation

Problem. Consider a linear operator L : R² → R²,

Find the matrix of L with respect to the basis v₁ = (3, 1), v₂ = (2, 1).

Let N be the desired matrix. Columns of N are coordinates of the vectors L(v₁) and L(v₂) w.r.t. the basis v₁, v₂.

Change of basis for a linear operator

Let L : V → V be a linear operator on a vector space V.

Let A be the matrix of L relative to a basis a₁, a₂ , . . . , an for V . Let B be the matrix of L relative to another basis b₁ , b₂ , . . . , bn for V .

Let U be the transition matrix from the basis a₁, a₂ , . . . , an to b₁ , b₂ , . . . , bn .

It follows that UA = BU .

Then A = U⁻¹BU and B = UAU ⁻¹.

Problem. Consider a linear operator L : R² → R²,

Find the matrix of L with respect to the basis v₁ = (3, 1), v₂ = (2, 1).

Let S be the matrix of L with respect to the standard basis, N be the matrix of L with respect to the basis v₁, v₂, and U be the transition matrix from v₁, v₂ to e₁, e₂.

Then N = U⁻¹SU .