MTH1004 Linear Algebra Practical Class 8

Question 1

Question

Consider the following transformations: (i) $T:{\mathbb{R}}^2\to {\mathbb{R}}^3$ $T\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-y\\ x-y\\ 2x-y\end{array}\right]$ (ii) $T:{\mathbb{R}}^3\to {\mathbb{R}}^3$ $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}x+2y\\ y-2z\\ x+y+z\end{array}\right]$ (iii) $T:{\mathbb{R}}^3\to {\mathbb{R}}^2$ $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}x-y+z\\ x+y-z\end{array}\right]$

Solution

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Question 2

Question

Show that these transformations are linear.

Solution

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Question 3

Question

Find a basis for the kernel and the range of the linear transformations (in Problem 1). What is their nullity and rank?

Solution

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Question 4

Question

Which of the linear transformations (in Problem 1) are one-to-one and which are onto? Is any of them an isomorphism? Justify your answer.

Solution

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