00:01
In this question we have been given a linear transformation defined from r2 to r2, okay, which is given by f of vector x equals to a times vector x, okay, where we have been given the matrix a as minus 2, minus 6, 9, 13, 5 and 8.
00:24
We need to find out the basis for kernel and the image of f so we have to find the basis of the kernel of f and the image of f so let us see how we are going to do this so first of all to find for the kernel we consider a x equals to zero vector a is given minus two minus six nine 13, 5, and 8, and x, i will take it as x1, x2, and this is equals to the zero vector, which is the column matrix with zero and trees.
01:13
So solving this, what i will get, i get minus 2x1 minus 6x2 equals to 0.
01:19
Then i get 9x1 plus 13x2 equals to 0 and 5x1 plus 8 x2 equals to 0.
01:29
Number of variable is less, number of equation is more so we get the trivial solution so this will imply that x1 x2 will be just zero only hence the kernel of f is going to contain only the element 0 comma 0 correct so this will be the basis only and now we have to calculate the basis for the image of f so what i'm going to get.
02:03
So first of all, we know that for r2, the basis are the set containing two element e1 and e2, where we know e1 is 1 comma 0 and e2, and e2 is 0 .1...