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Solving linear equations

Subspaces

Some sets of vectors are closed under adding and scaling: lines and planes through the origin, and the spaces a matrix carries with it. This topic finds the null space, writes every solution of a system, counts dimensions with a basis, and fits the four fundamental subspaces together.

Lessons on this topic

Animation: the plane x + y + z = 0 in a slowly turning 3-D view. V = (1, −1, 0) and w = (0, 1, −1) lie on it. Their sum (1, 0, −1) is on the plane. The stretched copy 2v = (2, −2, 0) is on the plane.Animation: the plane x + y + z = 0 in a slowly turning 3-D view. V = (1, −1, 0) and w = (0, 1, −1) lie on it. Their sum (1, 0, −1) is on the plane. The stretched copy 2v = (2, −2, 0) is on the plane.

Intermediate · 20 minYou are hereRead

Vector spaces and subspaces

Test whether a set of vectors is closed under adding and scaling, see why every subspace passes through zero, and meet the column space.

Animation: in a slowly turning 3-D view, a cloud of points and the line through the origin along (−1, −1, 1). The matrix A has moved every point to Ax: the cloud lies flat on the plane b3 = b1 + b2. Every point of the line is at the origin.Animation: in a slowly turning 3-D view, a cloud of points and the line through the origin along (−1, −1, 1). The matrix A has moved every point to Ax: the cloud lies flat on the plane b3 = b1 + b2. Every point of the line is at the origin.

Intermediate · 25 minYou are hereRead

The null space

Find every input a matrix sends to zero, by reducing to reduced row echelon form and building one special solution per free column.

Animation: in a slowly turning 3-D view of the inputs x, the solutions of Ax = (2, 3, 5). The null line, shifted to pass through (2, 3, 0). 5 points on it are marked, each with Ax = (2, 3, 5).Animation: in a slowly turning 3-D view of the inputs x, the solutions of Ax = (2, 3, 5). The null line, shifted to pass through (2, 3, 0). 5 points on it are marked, each with Ax = (2, 3, 5).

Intermediate · 20 minYou are hereRead

The complete solution

Write every solution of Ax = b as one particular solution plus the null space, and tell from the rank when there are none, one or infinitely many.

The point (5, 4) on the grid of v1 = (2, 1) and v2 = (1, 2). Its coordinates on that grid are (2, 1).The point (5, 4) on the grid of v1 = (2, 1) and v2 = (1, 2). Its coordinates on that grid are (2, 1).

Intermediate · 25 minYou are hereRead

Independence, basis and dimension

Draw a grid of your own with two arrows, read a point’s coordinates on it, and see why every basis of a space has the same size.

Two planes for the matrix [1, 2; 3, 6]. In the input plane the row line runs along (1, 2) and the null line along (−2, 1); in the output plane the column line along (1, 3) and the left null line along (−3, 1). The point (3, 1) is (1, 2) plus (2, −1) and lands on (5, 15).Two planes for the matrix [1, 2; 3, 6]. In the input plane the row line runs along (1, 2) and the null line along (−2, 1); in the output plane the column line along (1, 3) and the left null line along (−3, 1). The point (3, 1) is (1, 2) plus (2, −1) and lands on (5, 15).

Intermediate · 25 minYou are hereRead

The four fundamental subspaces

Follow a rank-one matrix from its input plane to its output plane, and meet the four subspaces every matrix carries, their sizes and their right angles.

Animation: a network of four nodes and five directed edges, 1 to 2, 2 to 3, 1 to 3, 1 to 4 and 3 to 4. A unit current circulates round the loop 1 to 3 to 4 to 1, edge values (0, 0, 1, −1, 1). Net current at the nodes (0, 0, 0, 0).Animation: a network of four nodes and five directed edges, 1 to 2, 2 to 3, 1 to 3, 1 to 4 and 3 to 4. A unit current circulates round the loop 1 to 3 to 4 to 1, edge values (0, 0, 1, −1, 1). Net current at the nodes (0, 0, 0, 0).

Advanced · 25 minYou are hereRead

Graphs and networks

Write a network as an incidence matrix, read potentials and currents as vectors, and find Kirchhoff’s laws in its four subspaces.

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