Posted: August 26th, 2021
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Linear Algebra
Q1. What a Linear Transformation Rn → Rm 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 → Rm 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 → Rm 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 → Rm Is Always The Multiplication By A Matrix A With M Rows And N Columns
All linear transformations T: Rn → Rmisa linear matrix transformation in a sense, as illustrated by the expression below.
= = x1 + x2 + =,… xn
Therefore, the function, T: Rn → Rm 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 → Rm
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 Rmsince 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 = x2,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.
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