91Ó°ÊÓ

Writing Explain why two similar matrices have the same rank.

let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation. Find the nullity of \(T\) and give a geometric description of the kernel and range of \(T\). $$ \operatorname{rank}(T)=1 $$

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 contraction represented by \(T(x, y)=(x, y / 2)\)

Finding the Inverse of a Linear Transformation In Exercises \(31-36,\) determine whether the linear transformation is invertible. If it is, find its inverse. $$ T(x, y)=(x+y, x-y) $$

Finding the Inverse of a Linear Transformation In Exercises \(31-36,\) determine whether the linear transformation is invertible. If it is, find its inverse. $$ T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}, x_{1}+x_{2}, x_{1}+x_{2}+x_{3}\right) $$

let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation. Find the nullity of \(T\) and give a geometric description of the kernel and range of \(T\). $$ \operatorname{rank}(T)=0 $$

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 expansion represented by \(T(x, y)=(x, 6 y)\)

Proof Prove that if \(A\) and \(B\) are similar matrices, then \(A^{T}\) and \(B^{T}\) are similar matrices.

let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation. Find the nullity of \(T\) and give a geometric description of the kernel and range of \(T\). $$ \operatorname{rank}(T)=3 $$

Finding the Inverse of a Linear Transformation In Exercises \(31-36,\) determine whether the linear transformation is invertible. If it is, find its inverse. $$ T\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=\left(x_{1}-2 x_{2}, x_{2}, x_{3}+x_{4}, x_{3}\right) $$