Linear Algebra

Posted: August 26th, 2021

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Linear Algebra

Q1. What a Linear Transformation Rⁿ → R^m Is

A linear transformation is a function that varies from one vector spaceV to another W and follows the basiclinear structure of each vector space. Therefore, it qualifies to be a function T when the linear transformation Rⁿ → R^m meets the following properties:

(i)T(x + y) = T(x) + T(y)

(ii) T(cx) = cT(x), where x, y∈Rⁿand c∈R

In the first (i) condition, the function T preserves additivity, whilepreservinga scalar multiplication in the second (ii) condition.

Q2. Examples Of Functions Rⁿ → R^m That Are Not Linear Transformation

As much as most transformation functions are linear, some are not, with examples as follows.

Cos(x + y) ≠ Cos(x) + Cos(y) or (2x)² ≠ 2(x²).

            For any transformation to be linear, the function T() = ; therefore, the non-linear transformation function will rule out the function,f(x) = x + 5.

            If we consider c = 0, then T() = (0 x ) = 0T() = . These two characteristics of a linear transformation could still be expressed as one concerning vectors u, v∈Rⁿas well as real numbers a, b ∈R.

T(a +b) = aT() + bT()

Q3. Explaining Why a Linear Transformation Rⁿ → R^m Is Always The Multiplication By A Matrix A With M Rows And N Columns

All linear transformations T: Rⁿ → R^misa linear matrix transformation in a sense, as illustrated by the expression below.

 =  = x₁ + x₂ + =,… x_n

Therefore, the function, T: Rⁿ → R^m is a multiplication of the matrix A with m rows and n columns because it is a linear combination of the vectors, as shown below.

 ,  ,…, 

Q4. Finding the Matrix Of A Linear Transformation Rⁿ → R^m

Supposing that A is a matrix of size m x n given by the following vector

            V = ∈RⁿTherefore defining the function T as T(v) = Av = A

It implies that the function T is a linear transformation from Rⁿ to R^msince it can be proved in respect to the characteristics of matrix multiplication u, v ∈Rⁿ and scalar c is expressively determined as follows;

T(u + v) = A(u + v) = A(u) + A(v) = T(u) + T(v) and

T(cu) = A(cu) = cAu = cT(u).

Q5. Examples

Example 1: Shear 2 X 2 matrix transformation where T: R² → R²

A =

For any horizontal vector  =

T() = A =  =  =

Therefore, function T is regarded as the identity in the horizontal vectors.

For vertical vectors like  = , The case is different, as expressed below.

T() = A =  =  = +

In this case, a vertical vector is usually moved perfectly horizontally through distance   times its length, as demonstrated using the diagram below.

Example 2: Scaling matrix transformation encompasses a 2 x 2 matrix,

WhereA =

For any vector, therefore,  =

T() = A =  =  = 

In the diagram below, the function T stretches horizontally as it contracts vertically.

Example 3: Reflection via a line, where we use the scenario in that

A =

T() = A =  = = 

In this scenario, the function T exchanges the two coordinates, thus reflecting via line x₁ = x_2,as illustrated below.

Example 4: Rotation is a linear example of matrix transformation,

Where A

T ( = A  =  =

When the horizontal unit vector is rotated c-clockwise through an angle . Likewise, the vertical unit vector  is rotated, as demonstrated below.