Matrix Representation of a Linear Transformation: Example Solved Problems

MATRIX REPRESENTATION OF A LINEAR TRANSFORMATION

WORKED EXAMPLES

Example 1

Let R³→R², defined by T (x, y, z) = (x+y, y + z), then β = {(‒1, 1, 1), (1, ‒1, 1), (1, 1, −1)}, and γ= {(1, 0), (0, 1)}. Then find [T].

Solution:

Let v₁ =(1, 1, 1), v₂ = (1, 1, 1), v₃ = (1, 1, 1), w₁ = (1, 0) and w₂ = (0, 1).

T(v₁)T(−1, 1, 1) = (−1 + 1, 1 + 1) = (0, 2)

T (v₂)=T(1, −1, 1) = (−1 − 1, − 1 + 1) = (0, 0)

T (v₃)=T(1, 1, −1)=(1+1, 1−1)=(2, 0)

(0,2)=0 (1,0)+2 (0, 1)

(0, 0) = 0 (1, 0)+0 (0, 1)

(2, 0) = 2 (1, 0)+0 (0, 1)

Example 2

Let T : R² → R³, defined by T (a₁, a₂) = (a₁+3α₂, 0, 2α₁−4a₂), then β = {(1, 0), (0, 1)} and γ= {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. Then find [T].

Solution:

Let v₁ = (1, 0), v₂ = (0, 1), w₁ = (1, 0, 0), w₂ = (0, 1, 0) and w₃ = (0, 0, 1)

T(v₁)=T(1, 0) = (1, 0, 2) = 1 (1, 0, 0) + 0 (0, 1, 0) + 2 (0, 0, 1)

= (1, 0, 0) + (0, 0, 0) + (0, 0, 2) = (1 +0 +0, 0+0+0,0 + 0 + 2) = (1, 0, 2)

T(v₂)=T(0, 1)=(3, 0, −4) = 3(1, 0, 0) + 0 (0, 1, 0) ‒ 4 (0, 0, 1)

= (3, 0, 0) + (0, 0, 0) + (0, 0, − 4) = (3+0 +0,0 +0+0,0 + 0−4) = (3, 0, −4)

Example 3

Let T: P₃(R)→P₂(R) be the linear transformation defined by T(f(x)) = f '(x). Let β and γ be the standard bases for P₃(R) and P₂(R) respectively. Then find [T].

Solution:

Given that T (f(x)) = f '(x).

 T(1)=0=0 (1) + 0 (x)+0(x²)

  T(x)=1=1 (1) + 0 (x)+0 (x²)

 T (x²) = 2x = 0 (1) + 2 (x) + 0 (x²)

 T (x³) = 3x² = 0 (1) + 0 (x) + 3 (x²)

Example 4

Let β & γ be the standard bases for Rⁿ and R^m respectively. Let T: Rⁿ→R^m. Compute [T]^γ_β for T: R² → R³ defined by T (a₁, a₂)=(2a₁−a₂, 3a₁+4a₂, α₁)

Solution:

Let us consider the linear transformation T: Rⁿ → R^m having standard bases β and γ respectively, for Rⁿ and R^m. The standard bases for R² are v₁ = (1, 0) & v₂ = (0, 1) and for R³ the bases are w₁ = (1, 0, 0), w₂ = (0, 1, 0) & w₃ = (0, 0, 1)

Let the transformation T: R² → R³

Defined transformation is T (a₁, a₂) = (2a₁−a₂, 3α₁ +4a₂, a₁)

 T(v₁) = T (1, 0) = (2 (1) − 0,3 (1) + 0, 1) = (2, 3, 1)

 T (v₂) = T(0, 1) = (2 (0) − 1, 3(0)+4(1), 0) = (− 1, 4, 0)

 The expression of [T] is

Example 5

Let T: R³ → R², defined by T (a₁, a₂, a₃) = (2a₁ +3a₂ − α₃, α₁ +a₃). Let β & γ be the standard bases for Rⁿ & R^m. Compute [T]_β^γ_.

Solution:

Given that T: R³ → R², T (a₁, a₂, a₃) = (2a₁ + 3a₂ – a₃, a₁ + a₃). Let v₁ =(1, 0) & v₂ = (0, 1) are the standard standard bases for R². Let w₁== (1, 0, 0), w₂ = (0, 1, 0) & w₃ = (0, 0, 1) are the standard bases for

Then T(w₁) = T (1, 0, 0) = (2 (1) +3 (0) −0,1+0) = (2, 1).

T (w₂) = T (0, 1, 0) = (2 (0) +3 (1) ‒ 0, 0+0) = (3, 0)

T(w₃) =  T (0, 0, 1) = (2 (0) + 3 (0) −1, 0+1) = (‒1, 1)

Example 6

Let T: R³ →R. Define T (a₁, a₂, α3) = (2a₁+a2−3a₃). Let β & γ be the standard bases for Rⁿ & R^m. Compute [T]_β^γ_.

Solution:

Let T: R³ → R

 T(a₁, a₂, a₃) = (2a₁ + a₂−3a₃)

Let v₁ = (1, 0, 0), v₂ = (0, 1, 0) & v₃ = (0, 0, 1) are the standard bases for R³.

T (v₁)=T(1, 0, 0) = (2 (1) + (0)−3(0))=2(0)

T (v₂) = T (0, 1, 0) = (2 (0) + (1) ‒ 3 (0)) = 1

and T (v₃)= T (0, 0, 1) = (2 (0) + (0)−3(1)) = −3

 [ T ] _β^γ_. = [2 1 − 3]

Example 7

Let T: Rⁿ→Rⁿ.

Define T (a₁, a₂, a_z ... a_n) = (a₁, α₁, a₁,... a₁). Compute [T].

Solution:

Given that T: Rⁿ →Rⁿ

T (a₁, a₂, a₃ ... a_n) = (a₁, a₁, a₁, a₁... a₁)

Let v₁ = (1,0,0... 0), v₂ = (0, 1, 0... O), v₃ = (0, 0, 1 ... 0)...  & v_n = (0, 0, 0 ... 1)

Then T(v₁) = T(1, 0, 0... 0) = (1, 1, 1, ... 1)

T (v₂) = T(0, 1, 0, ... 0) = (0, 0, 0, ... 0)

T (v₃) = T (0, 0, 1, 0 ... 0) = (0, 0, 0 ... 0)

T (v₄) T (0, 0, 0, 1, 0, ... 0) = (0, 0, 0... 0)

:

:

T (v_n) = T (0, 0, 0, ... 1) = (0, 0, 0 ... 0)

Example 8

Let T: R² → R³. Define T (a₁, a₂) = (a₁−a₂, a₁, 2a₁+a₂)

Let β be the standard bases for R² & γ be the standard bases for R³.

Given that γ = {(1, 1, 0), (0, 1, 1), (2, 2, 3)). Compute [T] _β^γ_.

Solution:

Given that T: R² → R³

T (a₁, a₂) = (a₁−a₂, a₁, 2a₁+ a₂)

Let β = {(1, 0), (0, 1)} be the standard bases for R²,

and γ = {(1, 0, 0), (0, 1, 0), (0, 0, 1) } be the standard bases for R³.

Here it is given that

 γ = { (1, 1, 0), (0, 1, 1), (2, 2, 3)} be the standard bases for R³.

Let v₁ = (1, 0), v₂ = (0, 1), w₁ = (1, 1, 0), w₂ = (0, 1, 1) & w₃ = (2, 2, 3)

T(v₁)=T(1, 0) = (1−0, 1, 2 (1) + 0) = (1, 1, 2).

T(v₂)=T(0, 1)=(0−1, 0, 2 (0) + 1) = (−1, 0, 1)

Given that γ = { (1, 1, 0), (0, 1, 1), (2, 2, 3)} be the standard bases for R³.

 T (v₁) = T (1, 0) = (1, 1, 2) = 1 (1, 1, 0) + 1 (0, 1, 1) +2 (2, 2, 3)

= (1, 1, 0) + (0, 1, 1) + (4, 4, 6)

= (5, 6, 7)

and T(v₂)=T(0, 1) = (− 1, 0, 1) = − 1 (1, 1, 0) + 0 (0, 1, 1) + 1 (2, 2, 3)

= (−1, −1, 0)+(0, 0, 0) + (2, 2, 3)

= (1, 1, 3)

Example 9

Let T: M_2×2 (R) → P₂ (R)

Define:  = (a+b) + (2d)x +bx² Let β = {} & γ = {1,x,x²}.

Compute [T]_β^γ.

Solution:

Given that T: M_2×2 (R) → P₂(R)

Let γ = {1, x, x²}

The coefficients of these linear combinations of the matrix is

Example 10

Let T: M_2×2 (F) → M_2×2(F).

Let α = {}, β = (1, x, x²) & γ = {1}.

Define T (A)=A^t, then compute [T]_α.

Solution:

Given that T (A)=A^t (or) A' where t or prime is transpose.

(ie) A= a_ij ⇒ A^t (or) A′ = a_ji

Example 11

Let T: P₂ (R) → M_2×2 (R). T[f(x)] =  and α = , β = (1,x,x²) & γ={1}.

Compute [T]^α_β.

Solution:

T[β]=[T]_β = 

Hence β = {1, x, x²}. so f(x)=(1, x, x²)

f '(x) = (0, 1, 2x), f ''(x) = (0, 0, 2)

  f ′ (0) = (0, 1, 0), 2f(1) = 2 (1, 1, 1) = (2, 2, 2), ƒ" (3) = (0, 0, 2)

Example 12

Let T: M_2×2(F) → F

Define: T (A) = Trace (A), a = {}

 Β ={1,x,x² } & γ=[1]. Compute [T]^γ_α.

Solution:

T: M_2×2(F) → F & T (A) = Trace (A) = a₁₁+ a₂₂ + a₃₃ + …. + a_nn

= [1 + 0, 0 + 0, 0 + 0, 0 + 1]

                   (by adding diagonal elements)

= [1, 0, 0, 1]

[ T ]^γ_β = (1, 0, 0, 1)