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

Determine a linear transformation function that would rotate the pink figure by

105^{\circ}

clockwise and would dilate it by a factor of 0.75 . The general form of the linear transformation should be

L (x, y) = (a x + b y, c x + d y)

. You must round all coefficients to four decimal places. The resulting figure should be the green shape.

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Finding the Kernel of a Transformation

(1 point) Let

{\vec{b}}_{1} = [\begin{matrix} - 1 \\ 3 \end{matrix}]

and

{\vec{b}}_{2} = [\begin{matrix} 3 \\ - 8 \end{matrix}]

. The set

B = {{\vec{b}}_{1}, {\vec{b}}_{2}}

is a basis for

R^{2}

. Let

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

be a linear transformation such that

T ({\vec{b}}_{1}) = 2 {\vec{b}}_{1} + 6 {\vec{b}}_{2}

and

T ({\vec{b}}_{2}) = 2 {\vec{b}}_{1} + 2 {\vec{b}}_{2}

. (a) The matrix of

T

relative to the basis

B

is (b) The matrix of

T

relative to the standard basis

E

for

R^{2}

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Projecting Using a Transformation

QUESIION7 Transforming a Vector from World Space to Local Space Choose one

\cdot 5

points A vector coordinate in world space is

\vec{r} = (1, 0, 2)

, calculate the vector coordinate in body frame coordinate (rigid body)

\vec{r} 0

if the orientation of the rigid body is 90 about the global z-axis. Note: use

\vec{r_{0}} = R^{t} \cdot \vec{r}

(\begin{array}{l} 0 \\ 1 \\ 2 \end{array})

(\begin{array}{l} 2 \\ 0 \\ 1 \end{array})

(\begin{matrix} 0 \\ - 1 \\ 2 \end{matrix})

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Finding the Pre-Image

Let the linear transformation

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

be defined by

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

. Find the pre-image of the vector

(6, 9, - 3)

under this transformation.

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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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