Column Space Calculator

The column space of a matrix $A$, often denoted as $\text{Col}(A)$, is the set of all possible linear combinations of its column vectors. To find the column space and its dimension (which equals the rank of the matrix), the calculator performs Gaussian elimination to reduce the matrix to its row echelon form or reduced row echelon form (RREF). The pivot columns in the RREF correspond directly to the linearly independent columns of the original matrix, which form the basis for the column space. The total number of these pivot columns gives the dimension of the column space.

Consider a 3x3 matrix $A$ where the columns are given by vector $v_1 = [1, 2, 1]^T$, $v_2 = [2, 4, 2]^T$, and $v_3 = [0, 1, 3]^T$. To find the column space, we arrange these vectors as columns in our matrix and apply row reduction. Through Gaussian elimination, we find that the first and third columns contain leading pivots, meaning columns 1 and 2 are linearly dependent (since column 2 is twice column 1). Thus, the basis for the column space consists of the vectors $[1, 2, 1]^T$ and $[0, 1, 3]^T$. The dimension of this column space is 2, indicating that the column vectors span a two-dimensional plane within three-dimensional space.

Always double-check your initial matrix entries to avoid sign errors or misplaced decimal points, as a single incorrect value can completely alter the resulting column space and rank. When working with large datasets or high-dimensional matrices, verify your row operations step-by-step or use this calculator to cross-reference manual homework solutions. Finally, remember that the dimension of the column space is always equal to the dimension of the row space, a fundamental theorem of linear algebra that serves as a useful sanity check for your results.