Thread: Rank and Dim of the image - relation?

1. Rank and Dim of the image - relation?

Hi - Need help in understanding the following

Let
X be a n-dim vector space
Y be a m-dim vector space
A be a mxn matrix

Thus, AX defined a linear transformation,T from X into Y i.e. T: X -> Y
Image of X under T, T(X) is a subspace of Y

I need help in understanding and thus formally proving two concepts:

1. Rank of A (which say is defined as dim of row space or dim of col space) = dim of T(X)

2. Why is dim(row space of A) = dim(col space of A)

Any pointers would be welcome. Thanks

2. Originally Posted by aman_cc
Hi - Need help in understanding the following

Let
X be a n-dim vector space
Y be a m-dim vector space
A be a mxn matrix

Thus, AX defined a linear transformation,T from X into Y i.e. T: X -> Y
Image of X under T, T(X) is a subspace of Y

I need help in understanding and thus formally proving two concepts:

1. Rank of A (which say is defined as dim of row space or dim of col space) = dim of T(X)

2. Why is dim(row space of A) = dim(col space of A)

Any pointers would be welcome. Thanks
For 1, Consider first a linear transformation $\displaystyle T:\mathbb{Re}^4 \rightarrow \mathbb{Re}^4$ associated to a 4 by 4 matrix A. Then, A has four columns. If three columns of them form a basis for a column space of A, the image of the corresponding matrix transformation spans 3-dimensional subspace of $\displaystyle \mathbb{Re}^4$.

Now consider a linear transformation $\displaystyle T:\mathbb{Re}^n \rightarrow \mathbb{Re}^m$ associated to a m by n matrix A. Then the column space of A consists of all vectors in $\displaystyle \mathbb{Re}^m$ that are images of at least one vectors in $\displaystyle \mathbb{Re}^n$ under matrix multiplication by A. If k columns of A form a basis for a column space of A, then the image of the corresponding matrix transformation spans k-dimensional subspace of $\displaystyle \mathbb{Re}^m$, where k is a positive integer less than or equal to m.

For 2, you can use the fact that if A is any matrix, then rank (A) = rank (A^T).