91Ó°ÊÓ

Find the standard matrix for the linear transformation \(T\). $$T\left(x_{1}, x_{2}, x_{3}\right)=(0,0,0)$$

Find all fixed points of the linear transformation. The vector \(\mathbf{v}\) is a fixed point of \(T\) if \(T(\mathbf{v})=\mathbf{v}\). A reflection in the \(x\) -axis

Find all fixed points of the linear transformation. The vector \(\mathbf{v}\) is a fixed point of \(T\) if \(T(\mathbf{v})=\mathbf{v}\). A reflection in the line \(y=-x\)

Sketch the image of the unit square with vertices at \((0,0),(1,0),(1,1),\) and (0,1) under the specified transformation. \(T\) is a reflection in the \(x\) -axis.

Sketch the image of the unit square with vertices at \((0,0),(1,0),(1,1),\) and (0,1) under the specified transformation. \(T\) is a reflection in the line \(y=x\).

Sketch the image of the unit square with vertices at \((0,0),(1,0),(1,1),\) and (0,1) under the specified transformation. \(T\) is the shear given by \(T(x, y)=(x, y+3 x)\).

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

Let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation. Use the given information to find the nullity of \(T\) and give a geometric description of the kernel and range of \(T\). \(T\) is the reflection through the \(y z\) -coordinate plane: \(T(x, y, z)=(-x, y, z)\)

Let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation. Use the given information to find the nullity of \(T\) and give a geometric description of the kernel and range of \(T\). \(T\) is the projection onto the vector \(\mathbf{v}=(1,2,2)\) \(T(x, y, z)=\frac{x+2 y+2 z}{9}(1,2,2)\)

Give a geometric description of the linear transformation defined by the elementary matrix. $$A=\left[\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}\right]$$