91Ó°ÊÓ

Sketch the image of the rectangle with vertices at \((0,0),(1,0),(1,2),\) and \((0,2)\) under the specified transformation. \(T\) is the shear represented by \(T(x, y)=(x+y, y)\)

Sketch the image of the rectangle with vertices at \((0,0),(1,0),(1,2),\) and \((0,2)\) under the specified transformation. \(T\) is the shear represented by \(T(x, y)=(x, y+2 x)\)

Calculus Define \(T: P_{2} \rightarrow R\) by \(T(p)=\int_{0}^{1} p(x) d x\) What is the kernel of \(T ?\)

Proof Let \(T: V \rightarrow W\) be a linear transformation. Prove that \(T\) is one-to-one if and only if the rank of \(T\) equals the dimension of \(V\).

Determine the matrix that produces the pair of rotations. Then find the image of the vector \((1,1,1)\) under these rotations. \(30^{\circ}\) about the \(z\) -axis and then \(60^{\circ}\) about the \(y\) -axis

Use the concept of a fixed point of a linear transformation \(T: V \rightarrow V .\) A vector \(\mathbf{u}\) is a fixed point when \(T(\mathbf{u})=\mathbf{u}\) (a) Prove that 0 is a fixed point of any linear transformation \(T: V \rightarrow V\). (b) Prove that the set of fixed points of a linear transformation \(T: V \rightarrow V\) is a subspace of \(V\). (c) Determine all fixed points of the linear transformation \(T: R^{2} \rightarrow R^{2}\) represented by \(T(x, y)=(x, 2 y)\). (d) Determineall fixed points of the linear transformation \(T: R^{2} \rightarrow R^{2}\) represented by \(T(x, y)=(y, x)\).

Prove that (a) the zero transformation and (b) the identity transformation are linear transformations.

Prove that \(T: V \rightarrow W\) is a linear transformation if and only if \(T(a \mathbf{u}+b \mathbf{v})=a T(\mathbf{u})+b T(\mathbf{v})\) for all vectors \(\mathbf{u}\) and \(\mathbf{v}\) and all scalars \(a\) and \(b\) Getting Started: This is an "if and only if" statement, so you need to prove the statement in both directions. To prove that \(T\) is a linear transformation, you need to show that the function satisfies the definition of a linear transformation. In the other direction, let \(T\) be a linear transformation. Use the definition and properties of a linear transformation to prove that \(T(a \mathbf{u}+b \mathbf{v})=a T(\mathbf{u})+b T(\mathbf{v})\). (i) Let \(T(a \mathbf{u}+b \mathbf{v})=a T(\mathbf{u})+b T(\mathbf{v}),\) Show that \(T\) preserves the properties of vector addition and scalar multiplication by choosing appropriate values of \(a\) and \(b\). (ii) To prove the statement in the other direction, assume that \(T\) is a linear transformation. Use the properties and definition of a linear transformation to show that \(T(a \mathbf{u}+b \mathbf{v})=a T(\mathbf{u})+b T(\mathbf{v})\).