Parts C D and E 3. For each of the following statements, determine whether it is true or false. If you think it is true, prove it.Otherwise, provide a counterexample to show that it is false.(a) A tall matrix A E Rmxn with m> n can never be onto.(b) A fat matrix A € Rmxn with m

Answered: (c) Similarity transformation always…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Parts C D and E

3. For each of the following statements, determine whether it is true or false. If you think it is true, prove it.
Otherwise, provide a counterexample to show that it is false.
(a) A tall matrix A E Rmxn with m> n can never be onto.
(b) A fat matrix A € Rmxn with m <n always has nontrivial nullspace, i.e., N(A) 2 {0}.
(c) Similarity transformation always preserves the range, i.e., for any A E Rxn and any nonsingular
TERnxn, R(A) = R(T-¹AT).
(d) For any A € Rmxn and any BE RnXP, R(AB) CR(A).
(e) For any A E Rmxn and any CE RPXm, N(A) CN(CA).

Expert Solution

Step 1 Part c

Similarity transformation always preserve the range. This statement is false.

For example, consider the linear transformations $T_{1} : ℝ^{2} \to ℝ^{2}$ and $T_{2} : ℝ^{2} \to ℝ^{2}$ defined by

$T_{1} (x, y) = (x, x) T_{2} (x, y) = (x, 0)$

The standard matrices corresponding to $T_{1}$ and $T_{2}$ are $[T_{1}] = A = [\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}]$ and $[T_{2}] = B = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$ .

To show that A and B are similar

The eigen values of A are 1 and 0 ( because trace(A)=1 and det(A)=0 ; also sum of eigen values of A = trace(A)=1 and product of eigen values of A = det(A) = 0. The two numbers whose sum is 1 and product is 0 are 1 and 0 ).

Hence the Jordan canonical form of A is $[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$ which is B. So matrices A and B are similar( a matrix is always similar to its Jordan canonical form).

But $R (A)$ is the straight line y = x and $R (B)$ is the Y axis. Hence $T_{1}$ and $T_{2}$ have different ranges.