Change of Basis Calculator

Inputs

Dimension (n) 2 3 4 5 6

Choose 2–6 for comfortable input.

Conversion mode B-coordinates → C-coordinates C-coordinates → B-coordinates Standard → B-coordinates Standard → C-coordinates B-coordinates → Standard C-coordinates → Standard

“Standard” means usual coordinates in ℝⁿ.

Basis B vectors (one per line) 1 0 0 0 1 0 0 0 1

Each line has n numbers. Example for n=3: 1 0 0

Basis C vectors (one per line) 1 1 0 0 1 1 1 0 1

Vectors are interpreted as columns when building matrices.

Vector (space or comma separated)

Enter n numbers matching the selected mode.

Submit Reset

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Example data table

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Formula used

• Coordinates in basis: if v is standard, then x_B = B⁻¹v and x_C = C⁻¹v.
• Change of basis: from B-coordinates to C-coordinates, x_C = (C⁻¹B)x_B.
• Reverse change: from C-coordinates to B-coordinates, x_B = (B⁻¹C)x_C.
• Back to standard: v = Bx_B or v = Cx_C.

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How to use this calculator

1. Select the dimension n that matches your vectors.
2. Choose a conversion mode between bases or standard form.
3. Enter n basis vectors for B and C, one per line.
4. Type the input vector as n numbers.
5. Press Submit. Results appear above this form.
6. Use CSV or PDF exports to save outputs.