# Image of a transformation

• Sep 24th 2009, 11:09 PM
noles2188
Image of a transformation
Describe the image of the transformation T(vector x) = A(vector x) geometrically (line, plane in R^2 or R^3)

; means a new row

A = [2 1 3 ; 3 4 2 ; 6 5 7]. I multiplied this matrix by the [x1 ; x2; x3]. I got x1[2 ; 3 ; 6] + x2[1 ; 4 ; 5] + x3[3 ; 2 ; 7]. Where do I go from here? I'm thinking it's a plane in R^3 but I don't know for sure.

Thanks
• Sep 25th 2009, 12:14 AM
NonCommAlg
Quote:

Originally Posted by noles2188
Describe the image of the transformation T(vector x) = A(vector x) geometrically (line, plane in R^2 or R^3)

; means a new row

A = [2 1 3 ; 3 4 2 ; 6 5 7]. I multiplied this matrix by the [x1 ; x2; x3]. I got x1[2 ; 3 ; 6] + x2[1 ; 4 ; 5] + x3[3 ; 2 ; 7]. Where do I go from here? I'm thinking it's a plane in R^3 but I don't know for sure.

Thanks

reduce A to echelon form or note that $2v_1 - v_2=v_3,$ where $v_j$ is the j-th column of A. thus the rank of A is 2 and therefore the image is a plane spanned by $v_1,v_2.$