I have watched 3 blue one brown Essence of Linear Algebra tutorials enter link description here. I want to ask one question. Let suppose we have a 3x4 transformation matrix as follows
A =
7 3 -9 8
6 2 -8 7
8 4 0 7
The basis of vector A are
{ (7,3,-9,8), (6, 2, -8, 7), (8, 4, 0, 7) }
And here is the vector which is to be transformed
B =
1 2 4 5
When we will multiply the matrix with vector it will take it from R4 --> R3. as
17 13 51
Please tell me, when we write the multiplication in terms of linear combination it is like that.
1 [ 7 6 8 ] + 2 [ 3 2 4] + 4 [ -9 8 0 ] + 5 [ 8 7 7]
But as the video of 3blue one brown suggested for R2 --> R2 transformation.
a( transformed vector i ) + b (transormed vector j) for vector to be transformed [a, b]
But in the case defined above it is streching its coloumns which are not vectors.
Please tell me the solution of this, or any wrong concept I have. Thanks
The columns of the matrix are vectors too, though in this case in $\mathbb R^3$ (not $\mathbb R^4$ like the rows).
It is a fact that the image of a linear transformation is always the span of the $\mathbf{columns}$ of any matrix representing it.