Question

In Linear algebra:

(x1-x2+x3+x4; 2x1-2x2+3x3+4x4; 3x1-3x2+4x3+5x4)

Find the basis for the image of T, Im(T).

Not sure how to find the image of a basis? I know how to find the kernel but not the basis? Anyone know how to do this.

Thanks in advance!

Answer 1

3

Answer 2

Find the matrix representation of the map T. Then find the basis of the column space of this matrix.

Answer 3

How do i find the matrix representation of the map T?

Thanks

Answer 4

can you find a matrix A so that Ax=(x1-x2+x3+x4; 2x1-2x2+3x3+4x4; 3x1-3x2+4x3+5x4) where x=(x1 x2 x3 x4) written as columnb.

Answer 5

Not sure if you can. would you start by putting it into row echlon form and then solving?

Answer 6

and then solve it*

Answer 7

to find the basis of column space you turn the matrix into echelon form and find out which column that has the leading 1. The same columns of the original matrix will be the basis of the columns space. It should be on the example of your textbook.

Answer 8

Ok thanks. Unfortunatley we only have lecture notes. no book :(
What do you mean by the column which has the leading 1? Is it the pivot columns.

Answer 9

Also , in the lecture notes. In the lecturers answer, he has a 3x4 matrix but we start with a 4x3 matrix which is why I am a bit confused?

Answer 10

yes, the matrix is of the size 3 x 4. The first row is (1 -1 1 1), second (2 -2 3 4) and the last row is ( 3 -3 4 5)

Answer 11

This is the answer that we were given:

Introducing standard basis feg4 i=1 (where e1 = (1; 0; 0; 0) and so
on) the image can be thought of as the vector space spanned by vectors
T(ei)'s. We can nd them from the denition of the linear mapping
T(e1) = (1; 1; 1); T(e2) = (-1; 0; 1); T(e3) = (1; 2; 3); T(e4) = (1;-1;-3)

Answer 12

From this he gets the matrix:

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

Answer 13

Do you understand that paragraph?

Answer 14

thanks for all of this help btw

Answer 15

@waterineyes What am i gonna learn from this ?

Answer 16

ok ashwin, if the matrix is written in that way then you are looking for the basis of the row space. Yes, you do row operation until you get matrix of echelon form. The basis that you are looking for are the rows of your echelon matrix that contain a leading 1.