2 suppose l r3 r3 is a linear transformation given defined...

2. Suppose L : R3 - R3 is a linear transformation given defined by (10 19 10 L= 6 13 8 v, 5 13 9 and let %= (a) Show that 9 is a basis of R3 (b) Write the matrix associated to L in the basis . (c) Interpret the linear transformation L geometrically d Discuss how the standard basis and the basis % are useful to describe the linear transformation L.

Transcript

00:01 Hello students, in the question we are provided with the linear transformation t from r3 to r3 defined by t of x comma y comma z is equal to minus 3 divided by 5 x1 plus 4 divided by 5 x3 comma x2, 4 divided by 5 x1 plus 3 divided by 5 x3.

00:36 For the first sub question, let us find t of 1 .0 .0 which is equal to minus 3 divided by 5 comma 0 .0 which can be written as minus 3 divided by 5 multiplied by 1 .00.

00:57 Plus 0 multiplied by 010 plus 4 divided by 5 multiplied by 001.

01:07 Similarly, t of 010 is equal to 010, which can be written as 0 multiplied by 1 00 plus 1 multiplied by 010 plus 1 multiplied by 010 plus 0 multiplied by 010 1.

01:29 Also, tf001 is equal to 4 by 5, 0, 3 by 5, which is equal to 4 divided by 5, 0 3 divided by 5, which is equal to 4 divided by 5 multiplied by 1 0 0 ,000 plus 0 multiplied by 010 plus 3 divided by 5 multiplied by 1 0 ,000 plus 3 divided by 5 multiplied by, 0 -0 -1.

02:10 Therefore our required matrix t s is equal to minus 3 divided by 5 0 4 divided by 5 transpose which is equal to minus 3 divided by 5 0 4 divided by 5 transpose which is equal to minus 3 divided by 5 0 4 divided by 5 0 1 4 divided by 5, 0, 3 divided by 5.

02:51 Now for the next question, we have to find t of 1 -0, 0, which is equal to minus 3 divided by 5 comma 0, 4 divided by 5, which can be written as 3 divided by 5 multiplied by 1 0, 2 plus 0 multiplied by 0, 0 ,000, plus 0 multiplied by 0.

03:24 1 -0 minus 2 divided by 5 multiplied by 2 -0 minus 1.

03:36 Similarly, t of 0 -1 -0 can be written as 0 multiplied by 102 plus 1 multiplied by 010 plus 0 multiplied by 0 -1 minus 1 and t of 0 -0 -1 1.

04:03 Can be written as 2 divided by 5 multiplied by 1 .02 plus 0 multiplied by 010 plus 1 divided by 5 multiplied by 2 0 minus 1.

04:21 Which implies t s comma b is equal to 3 divided by 5 comma 0 2 by 5 equal to 3 divided by 5 0 2 by 5 0 2 by 5.

04:45 2 by 5.

04:45 Divided by 5, 0, 1, 0, minus 2 divided by 5, 0, 1 divided by 5...