Analyze columns and find the generated output space. Inspect dependence, dimension, and rank in seconds. Solve classroom, homework, and study problems with structured output.
| Example Matrix | Pivot Columns | Basis for Image Space | Rank |
|---|---|---|---|
| [1 2 3], [0 1 1], [1 3 4] | 1, 2 | (1, 0, 1), (2, 1, 3) | 2 |
| [2 4], [1 2], [3 6] | 1 | (2, 1, 3) | 1 |
The image space of a matrix is the span of its column vectors.
Im(A) = span{a1, a2, ..., an}
To find a basis, row reduce the matrix to reduced row echelon form.
The pivot columns in the reduced matrix identify the matching columns in the original matrix.
Those original pivot columns form a basis for the image space.
Dimension of Image Space = Rank of the Matrix
The image space of a matrix describes every vector the matrix can produce. It is also called the column space. Each output vector comes from a linear combination of the matrix columns. This idea is central in linear algebra. It appears in transformations, systems of equations, data analysis, and geometry. When you find the image space, you learn how much of the target space the matrix can actually reach.
A matrix can have many columns, but not all of them add new direction. Some columns depend on others. The independent columns form a basis for the image space. To detect them, the calculator reduces the matrix to reduced row echelon form. The pivot columns in that reduced form show which original columns are independent. Those original columns become the basis vectors. Their count is the rank. That rank is also the dimension of the image space.
This calculator saves time and reduces manual errors. It checks the matrix size, reads each entry, performs row reduction, and reports the key outputs clearly. You can see the original matrix, the reduced row echelon form, the pivot columns, the non pivot columns, the basis vectors, and the final span statement. That makes it useful for homework, revision, lesson planning, and quick verification during problem solving. Export tools also help when you need a record.
If the rank is small, the image space is limited. The matrix does not cover many directions. If the rank is large, the matrix reaches more of the target space. This matters in modelling and computation. In applications, the image space can describe reachable states, valid outputs, compressed structure, or useful feature directions. By focusing on independent columns, you get a clean and efficient description. That is why image space analysis is a powerful step in matrix study.
The image space is the set of all output vectors created by the matrix. It is the span of the matrix columns. It is also called the column space.
The calculator row reduces the matrix to reduced row echelon form. It finds the pivot columns there. Then it takes the corresponding columns from the original matrix as the basis.
The reduced form helps locate independence, but the basis for the image space must come from the original column vectors. Those original pivot columns span the same image space.
Yes. The dimension of the image space equals the matrix rank. Both count the number of linearly independent columns in the matrix.
If every column depends on others, the rank becomes smaller than the total number of columns. The calculator will show only the independent columns in the basis.
Yes. You can enter any positive number of rows and columns. The method works for square matrices and rectangular matrices.
It shows where the pivots are located. Those pivot positions reveal which original columns form a basis for the image space.
It is useful in linear transformations, solving systems, vector space analysis, data modelling, and understanding how a matrix maps inputs into outputs.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.