Linear Transformations

Linear transformations are essentially functions that exist between a pair of vector spaces, maintaining the operations of scalar multiplication and vector addition. These transformations play a critical role in the realm of linear algebra, frequently symbolizing geometric transformations such as rotations, reflections, or scaling. These are particularly relevant in fields like computer graphics, physics, and engineering.

Proving a Transformation is Linear

Let

T : R^{2} \to R^{2}

be a linear transformation such that

T (x, y) = (3 x + 4 y, 2 x + 3 y)

. Prove that

T

is a linear transformation.

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Finding the Kernel of a Transformation

Let

T : R^{3} \to R^{3}

be a linear transformation defined by

T (x, y, z) = (2 x + y - 3 z, x - y + z, 3 x + 2 y - 4 z)

. Find the kernel of

T

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Projecting Using a Transformation

Given a vector

v = [2, 3]

and a transformation matrix

T = [[1, 0], [0, - 1]]

, what is the projection of

v

using the transformation matrix

T

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Finding the Pre-Image

Given the linear transformation

T : R^{2} \to R^{2}

defined by

T (x, y) = (2 x + y, x + 3 y)

. Find the pre-image of the point (5, 11) under the transformation T.

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Linear Transformations

Proving a Transformation is Linear

Finding the Kernel of a Transformation

Projecting Using a Transformation

Finding the Pre-Image

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