Mathematics II · Unit 2 · 4 hrs
Transformation
Exam-focused notes for Transformation (Mathematics II, MTH168): what the TU syllabus asks and how it has actually been tested, with 9 solved past questions from this unit.
What this unit covers
- Introduction to linear transformations
- the matrix of a linear Transformation
- Linear models in business, science, and engineering
Introduction to linear transformations
Define Linear Transformation with an Example
Let $A = \begin{bmatrix} 1 & -3 \ 3 & 5 \ -1 & 7 \end{bmatrix}$, $v = \begin{bmatrix} -2 \ 1 \end{bmatrix}$, $b = \begin{bmatrix} 3 \ 2 \ 1 \end{bmatrix}$, $x = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}$, $T(x) = Ax$
a. Find $T(v)$
b. Find $x \in \mathbb{R}^2$ whose image under $T$ is $b$ [10]
$$A = \begin{bmatrix} 1 & -3 \\ 3 & 5 \\ -1 & 7 \end{bmatrix}, \quad v = \begin{bmatrix} -2 \\ 1 \end{bmatrix}, \quad b = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}, \quad x = \begin{bmatrix} x1 \\ x2 \end{bmatrix}, \quad T(x) = Ax$$ --- A transformation $T:...
Full solved answer →Define linear transformation with an example. Let $$A = \begin{bmatrix} 1 & -3 \ 3 & 5 \ -1 & 7 \end{bmatrix}, \quad v = \begin{bmatrix} 2 \ -1 \end{bmatrix}, \quad b = \begin{bmatrix} 3 \ 2 \ 4 \end{bmatrix}, \quad x = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}$$
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]
A mapping $T: V \to W$ (where $V, W$ are vector spaces) is a linear transformation if for all vectors $\mathbf{u}, \mathbf{v}$ in $V$ and all scalars $c$: 1. $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ 2. $T(c\mathbf{v}) = c\,T(\mathbf{v})$...
Full solved answer →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(\mathb...
Full solved answer →Let $A = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}$ and define $T: \mathbb{R}^2 \to \mathbb{R}^2$ by $T(x) = Ax$. Find the image under $T$ of $u = \begin{bmatrix} 1 \ -3 \end{bmatrix}$ and $v = \begin{bmatrix} 1 \ 5 \end{bmatrix}$. [5]
$$A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \quad u = \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \quad v = \begin{bmatrix} 1 \\ 5 \end{bmatrix}$$ Transformation: $T(x) = Ax$, with $T:\mathbb{R}^2 \to \mathbb{R}^2$. --- $$T(u) = Au = \begin{bmatrix} 0 &...
Full solved answer →Define linear transformation with an example. Is a transformation T(x, y) = (3x + y, 5x + 7y, x+3y) linear? Justify. [5]
A transformation T: V → W is called a linear transformation if it satisfies the following two properties for all vectors u, v in the domain and all scalars c: 1. Additivity: T(u + v) = T(u) + T(v) 2. Homogeneity: T(cv) = c · T(v) The set ℝⁿ is called the do...
Full solved answer →the matrix of a linear Transformation
The columns of $I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$ are $e_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}$ and $e_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix}$. Suppose $T$ is a linear transformation from $\mathbb{R}^2$ into $\mathbb{R}^3$ such that $T(e_1) = \begin{bmatrix} 5 \ 1 \ -2 \end{bmatrix}$, $T(e_2) = \begin{bmatrix} 0 \ -1 \ 8 \end{bmatrix}$
Find a formula for the image of an arbitrary $x$ in $\mathbb{R}^2$. That is, find $T(x)$ for $x$ in $\mathbb{R}^2$. [2.5+2.5]
- Standard basis vectors: $e1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, $e2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ - $T: \mathbb{R}^2 \to \mathbb{R}^3$ linear - $T(e1) = \begin{bmatrix} 5 \\ 1 \\ -2 \end{bmatrix}$ - $T(e2) = \begin{bmatrix} 0 \\ -1 \\ 8 \en...
Full solved answer →Finding Standard Matrices of Linear Transformations
The problem has three separate parts (the codomain labels for parts 2 and 3 are printed as $\mathbb{R}^4$, but the descriptions are the standard $2\times2$ transformations; see notes below). Part 1: $T:\mathbb{R}^2\to\mathbb{R}^4$, with - $T(\mathbf{e}1)=(3...
Full solved answer →Let us define a linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ by $T(x) = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} -x_2 \ x_1 \end{bmatrix}$. Find the image under $T$ of $u = \begin{bmatrix} 4 \ 1 \end{bmatrix}$, $v = \begin{bmatrix} 2 \ 3 \end{bmatrix}$ and $u + v = \begin{bmatrix} 6 \ 4 \end{bmatrix}$. [5]
Linear transformation $T:\mathbb{R}^2 \to \mathbb{R}^2$: $$T(x) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} x1 \\ x2 \end{bmatrix} = \begin{bmatrix} -x2 \\ x1 \end{bmatrix}$$ Vectors: - $u = \begin{bmatrix} 4 \\ 1 \end{bmatrix}$ - $v = \b...
Full solved answer →define a transformation $T:\mathbb{R}^3 \to \mathbb{R}^2$ by $T(x) = Ax$ where
$$A = \begin{bmatrix} 1 & -5 & -7 \ -3 & 7 & 5 \end{bmatrix}, \quad u = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}, \quad b = \begin{bmatrix} -2 \ -2 \end{bmatrix}, \quad T(x) = Ax$$
a. Find $T(u)$
b. Find $x \in \mathbb{R}^3$ whose image under $T$ is $b$
c. Is $x$ unique?
[10]
$$A = \begin{bmatrix} 1 & -5 & -7 \\ -3 & 7 & 5 \end{bmatrix}, \quad u = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad b = \begin{bmatrix} -2 \\ -2 \end{bmatrix}, \quad T(x) = Ax$$ --- $$T(u) = Au = \begin{bmatrix} 1 & -5 & -7 \\ -3 & 7 & 5 \end{bmatrix}...
Full solved answer →Make Unit 2 stick
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