Vectors & Matrices
Vector & Matrix Operations (3D)
Vectors, A + B, Ax, AB, Aᵀ, A⁻¹ and permutation matrices step by step — and what each operation does in 3D.
Input
Computation
length: |result| = 11.6619
Move b so that its tail sits on the tip of a. The sum a + b runs from the tail of a to the tip of the moved b — the diagonal of the parallelogram spanned by a and b. Numerically: add the components.
- a
- b
- result
Input
Computation
Row by row: entry i is (row i of A) • x
Column by column: Ax is a linear combination of the columns of A
A is a (2×3)-matrix, so it maps vectors of ℝ³ to vectors of ℝ². Its j-th column is the image of the j-th basis vector: Ae₁ is column 1, and so on.
- x₁·a₁ (column 1 of A)
- x₂·a₂ (column 2 of A)
- x₃·a₃ (column 3 of A)
- Ax
Input
Computation
- Entries in the same position (row, column) are added — both matrices must have the same size.
- Column by column this is ordinary vector addition, as the picture shows.
- A + B = B + A (commutative) and (A + B) + C = A + (B + C) (associative); the zero matrix is the neutral element.
- aⱼ: columns of A (coloured by column)
- bⱼ attached at the tip
- cⱼ: columns of A + B
Input
A is (3×4), B is (4×2): the number of columns of A must equal the number of rows of B. AB is (3×2).
Computation
Click an entry of AB to see how it is computed.
c₂₁ = Σ a₂ⱼ·bⱼ₁ = (−2)·4 + (−7)·3 + 3·6 + 8·0 = −11
Entry (i, k) of AB is the dot product of row i of A with column k of B.
- In general AB ≠ BA, but the product is associative: (AB)C = A(BC).
- As transformations, AB means: first apply B, then A.
- E·A = A·E = A (identity); P·A swaps rows, A·P swaps columns.
The 3D picture is shown for square 2×2 or 3×3 matrices: there AB can be seen as a transformation — first B, then A. Try the presets “AB ≠ BA” or “3D: rotate → stretch”.
Identity matrix
Try it (Folie 8)
The result is A again, whichever side E is on.