Deriving the Column Space of a Matrix
To find the column space of a matrix, we essentially identify the building blocks of the given matrix—the set of linearly independent columns that can recreate every other column through addition and scaling.
Let’s work through a concrete example.
1. The Starting Matrix
Suppose you have the following matrix A:
[ 1 3 0 ]
[ 2 6 1 ]
[ 4 12 1 ]
By looking at it, you might notice that the second column is just the first column multiplied by three. T
his suggests that the second column is redundant and does not contribute to the “reach” of the matrix.
2. Row Reduction and Pivot Hunting
The first step is to simplify the matrix using Gaussian elimination to reach Row Echelon Form (REF).
This process reveals which columns are truly unique.
After subtracting multiples of the first row from the others, the matrix transforms into this:
[ (1) 3 0 ] <-- Pivot in Column 1
[ 0 0 (1) ] <-- Pivot in Column 3
[ 0 0 0 ]
The pivots (the first non-zero entries in each row) are located in the first and third columns. The second column has no pivot, confirming it is dependent on the others.
3. Selecting the Basis
To define the column space, you need a basis. A crucial rule in linear algebra is that you must select the columns from the original matrix, not the simplified one. Since our pivots were in columns 1 and 3, we grab those from the original matrix A:
Basis = { [1, 2, 4], [0, 1, 1] }
In linear algebra, the number of vectors in a basis defines the dimension of the space they create (the span).
4. Defining Col(A)
The column space, Col(A), is formally defined as the “span” of these basis vectors. This means Col(A) is the set of all possible vectors you can create by multiplying these basis vectors by any constants and adding them together.
In this specific case, because we have two basis vectors, the column space forms a 2D plane sitting inside 3D space.
Even though the vectors [1, 2, 4] and [0, 1, 1] live in 3D, there are only two of them. One can multiply them by any number and add them together all day long, but we will only ever be able to "draw" a flat surface using those two directions. We lack a third independent direction to leave that surface and fill the rest of the 3D room.
5. Conclusion and Rank
The number of vectors in the basis determines the Rank of the matrix. Here, the Rank is 2. This tells you that any vector you try to solve for (the “b” in Ax = b) must land exactly on that 2D plane to be solvable. If a vector points anywhere outside that plane, the system has no solution because the columns of the matrix simply cannot reach that area of space.