Consider the linear transformation T Rrightarrow RGiven that

Consider the linear transformation T: R^rightarrow R^Given that T[(1 0 0)] = (1 0), T[(0 1 1)] = (0 1), T[(-1 2 2)] = (-1 2), T (0 0 1) = (2 1) find T[(2 3 4)]

Solution

We know that a llinear transformation preserves vector addition and scalar multiplication. Hence for determining T(2,3,4)^T , we will have to determine (2,3,4)^T as a linear combination of (1,0,0)^T,(0,1,1)^T, (-1,2,2)^T and (0,0,1)^T . Let A be the matrix with these vectors and the vector (2,3,4)^T as columns.Then we have A =

1

0

-1

0

2

0

1

2

0

3

0

1

2

1

4

We will reduce A to its RREF as under:

Add -1 times the 2nd row to the 3rd row

Then the RREF of A is

1

0

-1

0

2

0

1

2

0

3

0

0

0

1

1

Now, it is apparent that (2,3,4)^T = 2(1,0,0)^T +3(0,1,1)^T +4(0,0,1)^T . Hence, T(2,3,4)^T =2T(1,0,0)^T +3T(0,1,1)^T +4T(0,0,1)^T = 2(1,0)^T +3(0,1)^T+4(2,1)^T = (2,0)^T+(0,3)^T+(8,4)^T = (10,7)^T.

Note: The image is very faint. This may cause some incorrect inputting of data. Please make changes if required.

1	0	-1	0	2
0	1	2	0	3
0	1	2	1	4