Algebra - Linear Transformation

Algebra - Linear Transformation

• Functions: A function maps each element of a set (domain) to exactly one element in another set (codomain).
• Univariate Function: A function of one variable, like f(x) = x².
• Bivariate Function: A function of two variables, like f(x₁, x₂)= 2x₁ + 3x₂.
• Multivariate Function: A function that takes more than two variables, like f(x₁, x₂, x₃) = 2x₁ + 3x₂ − x₁x₂ + 4x₁ + 5x₂.
• Polynomial Function: A function that can be expressed as a sum or difference of terms consisting of constants and variables raised to non-negative integer powers, like f(x, y) = 2x³ + 3x³ − x₁x₂ + 4x₁ + 5x₂.
• One-to-One (Injective): A function where each output corresponds to exactly one input.
• Onto (Surjective): A function where every element in the codomain has at least one corresponding input.
• Bijective: A function that is both one-to-one and onto.

Transformation

• Definition: A transformation is a function from one vector space to another.
• Domain: The set of input vectors.
• Codomain: The set of possible output vectors.
• Image: The result of applying a transformation to a specific input vector.
• Range: The set of all possible output vectors.

Matrix Transformation

• Definition: A matrix transformation is a transformation defined by multiplying an input vector by a matrix.
• Range: The range of a matrix transformation is the column space of the matrix.
• Linear Combination: Outputs are linear combinations of the matrix's columns.

Projection onto the xy-plane

• Matrix:

[ 1  0  0]
[ 0  1  0]
[ 0  0  0]

• Projects a vector onto the x-y plane, effectively setting the z-component to zero.

Reflection

• Matrix for reflection across the xy-plane:

[ -1  0  0]
[  0  1  0]
[  0  0  1]

• Reflections across the xy-plane negate the x-component.

Dilation

• Matrix for a dilation of 1.5 in both x and y directions:

[ 1.5  0  0]
[  0  1.5  0]
[  0  0  1]

• Multiplies the x and y components by 1.5, enlarging the image.

Rotation

• Matrix for a 90-degree counterclockwise rotation:

[ 0  -1  0]
[ 1   0  0]
[ 0   0  1]

Questions (Matrix Transformation)

• Find T(u): Calculates the transformed vector u.
• Find a vector v: Identifies a vector v mapped to b.
• w exists: Determines if a vector w can be expressed as a transformation of multiple vectors.
• Not in range: Finds a vector not within the transformation's possible outputs.

One-to-One Transformation

• Definition: A transformation where different inputs produce different outputs.
• Properties: The equation Ax = b has at most one solution (a consistent or inconsistent system). Also columns of A are linearly independent and the range of T has dimension n.
• Examples: Transformations with the identity matrix.

Onto Transformation

• Definition: A transformation where every element in the codomain is a possible output.
• Properties: The equation Ax =b is consistent for all b in Rm. This means the columns of the matrix A span Rm. Also, the range of T has dimension m.

Linear Transformation

• Definition: A transformation that satisfies two properties for all vectors x, y and scalars r, the transformation T is linear if: T(x+y) = T(x) + T(y) and T(rx) = rT(x).
• Matrix Transformation: Every matrix transformation is a linear transformation, making matrix transformations a special set of linear transformations..

Non-Linear Transformation

• Examples: Examples of transformations that do not satisfy the properties of linear transformations, such as T₁(x, y) = [x/y], T₂(x, y) = xy/y, and T₃(x, y) = [2x+1, x–2y].

Composition of Linear Transformations

• Definition: The composition of two linear transformations is a new transformation where the first transformation's output is the input for the second. The result is a linear transformation. Examples using matrix examples.