Determine if the following functions \(T{:}\ \mathbb R^2 \rightarrow \mathbb R^2\) are one-to-one and/or onto.
Determine if the following functions \(T{:}\ \mathbb R^2 \rightarrow \mathbb R^2\) are one-to-one and/or onto.
Let \(D\) be a square with vertices \((0,0),(1,1),(2,0),(1,-1)\) and \(D^*\) be a parallelogram with vertices \((0,0),(1,2),(2,1),(1,-1)\). Find a linear map \(T\) taking \(D^*\) onto \(D\).
Let \(D\) be a parallelogram with vertices \((0,0),(-1,3),(-2,0),(-1,-3)\). Let \(D^* = [0,1] \times [0,1]\). Find a linear map \(T\) such that \(T(D^*)=D\).
Let \(S^* = (0,1] \times [0, 2 \pi)\) and define \(T (r, \theta) = ( r \cos \theta, r \sin \theta)\). Determine the image set \(S\). Show that \(T\) is one-to-one on \(S^*\).
Define \[ T ( x^* , y^*) = \left( \frac{x^* - y^*}{\sqrt{2}}, \frac{x^* + y^*}{\sqrt{2}} \right). \] Show that \(T\) rotates the unit square, \(D^* = [0,1] \times [0,1]\).
Let \(D^* = [0,1] \times [0,1]\) and define \(T\) on \(D^*\) by \(T (u, v) = ( - u^2 + 4u,v)\). Find the image \(D\). Is \(T\) one-to-one?
Let \(D^*\) be the parallelogram bounded by the lines \(y = 3x - 4, y = 3x, y = {\textstyle \frac{1}{2}}x\), and \(y= {\textstyle \frac{1}{2}} (x+4)\). Let \(D = [0,1] \times [0,1]\). Find a \(T\) such that \(D\) is the image of \(D^*\) under \(T\).
Let \(D^* = [0,1] \times [0,1]\) and define \(T\) on \(D^*\) by \(T (x^*, y^*) = (x^*y^*, x^*)\). Determine the image set \(D\). Is \(T\) one-to-one? If not, can we eliminate some subset of \(D^*\) so that on the remainder \(T\) is one-to-one?
Let \(D^*\) be the parallelogram with vertices at \((-1,3), (0,0), (2,-1)\), and \((1,2)\), and \(D\) be the rectangle \(D= [0,1] \times [0,1]\). Find a \(T\) such that \(D\) is the image set of \(D^*\) under \(T\).
Let \(T \colon\, {\mathbb R}^3 \to {\mathbb R}^3\) be the spherical coordinate mapping defined by \((\rho, \phi, \theta) \mapsto (x,y,z)\), where \[ x = \rho \sin \phi \cos \theta, y = \rho \sin \phi \sin \theta, z = \rho \cos \phi. \]
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Let \(D^*\) be the set of points \((\rho, \phi, \theta)\) such that \(\phi \in [0, \pi], \theta \in [0, 2 \pi], \rho \in [0,1]\). Find \(D = T ( D^*)\). Is \(T\) one-to-one? If not, can we eliminate some subset of \(D^*\) so that, on the remainder, \(T\) will be one-to-one?
In Exercises 12 and 13, let \(T ( {\bf x}) = A {\bf x}\), where A is a \(2 \times 2\) matrix.
Show that \(T\) is one-to-one if and only if the determinant of \(A\) is not zero.
Show that det \(A \ne 0\) if and only if \(T\) is onto.
Suppose \(T \colon\, {\mathbb R}^2 \to {\mathbb R}^2\) is linear and is given by \(T ({\bf x}) = A {\bf x}\), where \(A\) is a \(2 \times 2\) matrix. Show that if det \(A \ne 0\), then \(T\) takes parallelograms onto parallelograms. [HINT: The general parallelogram in \({\mathbb R}^2\) can be described by the set of points \({\bf q} = {\bf p} + \lambda {\bf v}+ \mu {\bf w}\) for \(\lambda, \mu \in (0,1)\) where \({\bf p}, {\bf v}, {\bf w}\) are vectors in \({\mathbb R}^2\) with \({\bf v}\) not a scalar multiple of \({\bf w}\).]
A map \(T: \mathbb R^2 \rightarrow \mathbb R^2\) is called affine if \(T(\bf{x}) = A\bf{x} + \bf{v},\) where \(A\) is a \(2 \times 2\) matrix, and \(\bf{v}\) is a fixed vector in \(\mathbb R^2\). Show that Exercises 12, 13, and 14 hold for \(T\).
Suppose \(T \colon\, {\mathbb R}^2 \to {\mathbb R}^2\) is as in Exercise 14 and that \(T ( P^*) = P\) is a parallelogram. Show that \(P^*\) is a parallelogram.
Show that \(T\) is not one-to-one.