Updated Linear Algebra & Embedding Tool

Vector Dimension Calculator

Compute the dimension of the span of vectors, rank and a basis using linear algebra, or analyze embedding vectors by their dimension, norm, mean, variance and sparsity. Paste vectors, click calculate, and get instant, readable results for both math and AI use cases.

Vector Space Dimension Rank & Basis Embedding Dimension Norm & Statistics

Calculate Vector Space Dimension or Embedding Dimension

This Vector Dimension Calculator has two modes. The first tab treats each row as a vector and computes the dimension of the span, the rank, a basis and a row-reduced echelon form. The second tab takes a single embedding-like vector and reports its dimension, norm, basic statistics and sparsity, which is useful when working with modern AI embedding models.

In the first tab, paste multiple vectors line by line. In the second tab, paste a single long vector. Both tabs accept separators such as spaces or commas and ignore blank lines automatically.

Use spaces or commas between components. Each non-empty line is treated as one vector. All vectors must have the same length (same number of components).

The calculator builds a matrix from your vectors and uses Gaussian elimination to compute the rank and a basis of the row space. It reports the dimension of the span, a linearly independent subset of your vectors and a row-reduced echelon form for reference.

Paste a single vector. Use commas, spaces or line breaks as separators. Non-numeric entries are ignored.

This mode is helpful for inspecting embedding vectors from AI models. It reports the number of dimensions, magnitude (L2 norm), basic statistics (min, max, mean, variance, standard deviation) and an approximate sparsity level based on values close to zero.

Vector Dimension Calculator – From Linear Algebra to Embeddings

The Vector Dimension Calculator on MyTimeCalculator bridges the gap between classical linear algebra and modern embedding-based AI systems. In one tool, you can compute the dimension of the span of several vectors, determine linear independence, extract a basis and inspect the row-reduced echelon form. In another tab, you can analyze a single embedding vector and see its dimension, norm and summary statistics.

This is useful both for students learning about vector spaces and for practitioners working with high-dimensional embeddings in search, recommendation and retrieval-augmented generation (RAG) systems.

1. Vector Space Dimension and Rank (Linear Algebra Mode)

In the Vector Space Dimension (Linear Algebra) tab, each row of the textarea is treated as a vector in ℝⁿ. The calculator builds a matrix where each vector becomes a row and then applies Gaussian elimination to compute:

  • Rank: The number of linearly independent rows, which equals the dimension of the row space.
  • Dimension of the span: The dimension of the vector space spanned by your input vectors.
  • Linear independence: Whether all vectors are independent or some are linear combinations of others.
  • Basis: A subset of your vectors that forms a basis for the span.
  • Row-reduced echelon form (RREF): A cleaned-up matrix for visual inspection of pivot positions.

From a teaching perspective, this helps students see how rank and dimension relate to independence. From a practical standpoint, it can be used to quickly test whether feature vectors or signals carry redundant information.

2. Embedding Vector Dimension and Statistics (Embedding Mode)

In the Embedding Vector Dimension tab, you paste a single high-dimensional vector similar to those produced by modern embedding models. The calculator reports:

  • Dimension (length): The number of components in the vector.
  • L2 norm and L1 norm: Measures of magnitude that are often used in similarity calculations.
  • Minimum, maximum and mean: Basic distribution information for the components.
  • Variance and standard deviation: How spread out the components are.
  • Sparsity estimate: The proportion of components that are effectively zero (within a small tolerance).

These diagnostics are handy when comparing different models, checking if normalization steps worked correctly, or understanding how “dense” or “sparse” a given embedding representation is in practice.

3. How to Use the Vector Dimension Calculator

  1. Choose the tab that matches your goal. Use the linear algebra tab for multiple vectors and the embedding tab for a single long vector.
  2. Paste your vectors. In the linear algebra tab, put one vector per line. In the embedding tab, put all components of a single vector separated by spaces, commas or line breaks.
  3. Check that the parsing assumptions are correct. All vectors in the linear algebra tab must have the same number of components; the calculator will warn you if they do not.
  4. Click the calculate button. The results section shows key values in cards, and the linear algebra tab also displays an RREF matrix in a clean table format.
  5. Interpret the output. Dimension and rank tell you how many independent directions your vectors span. Norms and variance tell you about the scale and spread of embedding vectors.
  6. Experiment by modifying the input. Add or remove vectors to see how the dimension changes, or rescale components in an embedding to observe how it affects norms and statistics.

4. Practical Applications

  • Linear algebra and coursework: Quickly verify textbook exercises about rank, dimension and linear independence using actual numeric calculations.
  • Feature engineering: Test whether newly added features are likely to be linearly redundant with existing ones.
  • AI embeddings and RAG: Inspect the dimension and statistical properties of embeddings from different models, or verify that vectors have the expected dimension when integrating APIs.
  • Model debugging: Spot anomalies in embeddings such as extremely large norms or unexpected sparsity patterns that may indicate preprocessing or scaling issues.

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Vector Dimension Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about vector space dimension, rank, embedding dimension and how to interpret the outputs of this calculator.

The dimension of the span is the number of linearly independent vectors needed to generate all vectors in the set through linear combinations. It equals the rank of the matrix formed by placing the vectors as rows (or columns). If you have five vectors but the calculator reports dimension three, it means that only three of them are truly independent, while the others can be expressed in terms of those three.

The calculator performs Gaussian elimination and keeps track of pivot rows. The input vectors corresponding to these pivot rows form a linearly independent subset that spans the same space as the full set. This subset is reported as a basis. Different elimination paths may give different-looking bases, but all valid bases have the same number of vectors and span the same subspace.

For embedding vectors, the raw dimension (for example 768 or 1024) is only part of the story. The norm shows how large the vector is overall, while sparsity, mean and variance give a quick sense of how the components are distributed. These diagnostics are helpful when comparing models, debugging preprocessing or verifying that normalization steps behave as expected in your pipeline.

Linear dependence means at least one vector in your set can be written as a linear combination of the others. In practice, this indicates redundancy: some vectors do not add any new direction to the span. The calculator reports how many independent vectors you have and identifies a basis, which is a minimal set of vectors that still spans the same subspace without redundancy.

This online implementation is designed for real-valued vectors and expects each component to be a real number in decimal form. Complex numbers are not supported directly. If you need to work with complex vectors, you would typically separate real and imaginary parts or use a symbolic math package that handles complex arithmetic explicitly.

The calculator uses standard floating-point arithmetic in the browser, which can introduce tiny rounding errors. As a result, entries that should be exactly zero may appear as very small numbers such as 1e-12. The tool uses a numerical tolerance when deciding whether a pivot is non-zero, so the computed rank and dimension are robust, even if the displayed RREF has small rounding artifacts.

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