Understanding Linear Algebra Basis Results
What a Basis Means
A basis is a smallest useful set of vectors. It spans a space without repeating direction. This calculator checks those ideas with row reduction. It accepts a matrix or a list of vectors. Then it finds pivots, rank, row space, column space, null space, and independent input vectors.
Why Row Reduction Helps
The main step is reduced row echelon form. Each pivot marks a leading variable or a leading vector. Pivot columns from the original matrix form a basis for the column space. Nonzero rows from the reduced matrix form a basis for the row space. Free variables form the null space basis.
Choosing the Right Input Mode
You can enter vectors as rows or as columns. This is useful for homework, engineering notes, data checks, and abstract algebra practice. The orientation setting tells the tool how your vectors are arranged. If each line is a vector, choose row mode. If each column is a vector, choose column mode.
Testing a Target Vector
The target vector option tests span membership. The calculator solves a linear system with the target as the right side. A consistent system means the target lies in the span. The coordinate output gives one valid combination. Dependent sets may have many combinations, so free variables are set to zero.
Rank and Nullity
Rank is the number of pivots. Nullity is the number of columns minus rank. A full column rank matrix has independent columns. A square matrix with full rank has a column basis for the whole coordinate space. A zero matrix has no nonzero pivot direction.
Reading the Output
Results are shown in a clear order. First, the summary appears. Next, pivot columns and basis vectors appear. Then the reduced matrix and null space basis appear. The export buttons save the same work for records.
Accuracy Tips
Use a small tolerance for clean integer data. Use a larger tolerance for rounded decimal data. The precision field controls display only. It does not change the internal row operations. Fractions are approximated for readable answers.
Study Value
This tool is educational. It helps verify hand work and explain structure. It does not replace a proof. Always check the vector arrangement, units, and expected dimension before using final answers.
For best results, keep every row length equal. Separate numbers with spaces, commas, tabs, or semicolons before pressing calculate once for review.