Understanding row and column space

[FourierX]

Understading row and column space

Homework Statement

I am having hard time trying to understand row and column space. Can anyone simplify the meanings of them so that i can visualize them well.

Homework Equations

dimension of row space = rank ? How? Why?

The Attempt at a Solution

Understanding other vectors

 

[Mark44]

Let's look at a concrete example. Suppose A is a 2 x 3 matrix, shown here:
[1 3 0]
[2 5 1]

Each of the two rows can be considered to be a vector in R^3, and each of the three columns can be considered to be a vector in R^2.

The row space of this matrix is the set of all linear combinations of the row vectors, or
c_1(1, 3, 0) + c_2(2, 5, 1).

Similarly, the column space is the set of all linear combinations of the column vectors, or
d_1(1, 2)^T + d_2(3, 5)^T + d_3(0, 1)^T.

For this matrix, what is the dimension of the row space? the column space? What is the rank of this matrix?

Now suppose that A is an m x n matrix, meaning it has m rows and n columns. Each of the m rows can be considered a vector with n components (because there are n columns). Each of the n columns can be considered a vector with m components. Just as in the first example, the row space of this matrix is the set of all linear combinations or the row vectors, and the column space of this matrix is the set of all linear combinations or the column vectors.

 

[FourierX]

Understading row and column space

Thanks

on visualizing them in the graph or say pictorially, how would they look?

 

[Mark44]

The two row vectors are a basis for a plane in R^3. The three column vectors span R^2 but aren't a basis -- there are too many vectors for a basis.