Mathematics II207810 marksNumerical: linear transformation computationIntroduction to linear transformations
Define linear transformation with an example. Let A = beginbmatrix 1 & -3 3 & 5 -1 & 7 endbmatrix, v = beginbmatrix 2 -1 endbmatrix, b = beginbmatrix 3 2 4 endbmatrix, x = beginbmatrix x1 x2 endbmatrix and define a transformation T: mathbbR2 to mathbbR3 by T(x) = Ax then a. find T(v) b. find x in ma
Define linear transformation with an example. Let $A = \begin{bmatrix} 1 & -3 \ 3 & 5 \ -1 & 7 \end{bmatrix}$, $v = \begin{bmatrix} 2 \ -1 \end{bmatrix}$, $b = \begin{bmatrix} 3 \ 2 \ 4 \end{bmatrix}$, $x = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}$ and define a transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$ by $T(x) = Ax$ then
a. find $T(v)$
b. find $x \in \mathbb{R}^2$ whose image under $T$ is $b$ [10+0]
A transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation if for all vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ and all scalars $c$: 1. $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ 2. $T(c\mathbf{u}) = c,T(\mathbf{u})$ Example: For any $m \times n$ mat...