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Linear Algebra With Applications
Found in: Page 199
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

True or False.

The linear transformation T(M)=[1236]M from R2×2 to R2×2 have rank 1.

The given statement is False.

See the step by step solution

Step by Step Solution

Step 1: Determine the matrix B .

Consider the function T(M)=[1236]M from R2×2 to R2×2 .

As localid="1659426855825" 1000,0100,0010,0001is the basis element of R2×2, the matrix B is defined as follows.

T1000=12361000 =1030 =1000+30010


The image of T at point 0100 is defined as follows.

T0100=12360100 =0103 =0100+30001 =0.u1+1.u2+0.u3+3.u4

The image of T at point 0010 is defined as follows.

T0010=12360010 =2060 =21000+60010 =2.u1+0.u2+6.u3+0.u4

The image of T at point 0001 is defined as follows.

T0001=12360001 =0206 =20100+60001 T0001 =0.u1+2.u2+0.u3+6.u4

Therefore, the matrix B= 1020010230600306.

Step 2: Determine the rank of T .

As two rows are linearly dependent implies rank (T)=2

Hence, the statement is false.

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