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Inverse matrices, column space and null space

Explains linear systems geometrically through transformations, showing how inverses, rank, column space, and null space describe their solutions.

3Blue1Brown⏱ 12 minOpen on YouTube ↗

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What the lecture covers

A linear system can be written as Ax = v, where A represents a linear transformation. Solving the system means finding an input vector x that the transformation sends to v. If A has a nonzero determinant, it does not collapse space and has an inverse transformation. Applying that inverse to v gives the unique solution. If the determinant is zero, the transformation squashes space into a lower-dimensional set, so no inverse exists; a solution is possible only when v lies in the transformation’s output.

Rank measures the dimension of that output, which is the column space: the span of the columns of A. A matrix is full rank when its rank equals its number of columns. When a transformation collapses dimensions, multiple inputs may map to zero; all such inputs form the null space, also called the kernel. For a system with v equal to zero, the null space contains every solution. Together, the inverse, column space, and null space describe whether solutions exist and what they can look like.

Key ideas

A linear system can be packaged as Ax = v, where solving it means finding an input that the transformation A maps to v.
When A has a nonzero determinant, its inverse reverses the transformation and yields the unique solution x = A⁻¹v.
When a transformation collapses space, it has no inverse, though a solution can still exist if v lies in its output.
Rank is the number of dimensions in the transformation’s output.
The column space is the set of possible outputs, equal to the span of the matrix’s columns.
The null space, or kernel, consists of all inputs that the matrix maps to zero.

Sample questions

In a mathematical model, several unknown quantities are linked by multiple equations. Which description captures what makes this a system of equations?

AA single known value represented in several different ways
BA list of equations that contains no unknown quantities
CA collection of unknown variables together with equations that relate them
DA set of variables considered without any relationships between them

Show answer

Correct answer: C. A system combines unknown quantities with equations expressing relationships among those quantities.

For a matrix A viewed as a linear transformation and a target vector v, what does it mean for an input vector x to solve Ax = v?

AThe transformation leaves every possible input vector unchanged.
BApplying the transformation to x produces the target vector v.
CThe input vector x must be the zero vector.
DThe target vector v is transformed into the input vector x by A.

Show answer

Correct answer: B. A solution is an input whose image under the transformation equals the specified target.

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