Is it in the Span?

Linear Algebra (Matrices) · Axiom Academy

Reduce the span question to a linear system, row-reduce, and read off the verdict. Working in , decide whether lies in . If it does, give the weights c_1, c_2, c_3 with . The geometry behind the question Because , the three vectors are coplanar — their span is a plane . The target points off that plane, so no combination of them can reach it. You turned a span question into a row-reduction and read the verdict straight off the matrix. Span means solvable: exactly when has at least one solution. Vectors as columns: build and row-reduce. The contradiction row: a row with means 0=c — the system is inconsistent and is not in the span. This problem: reduction gave , so . Why, geometrically: , so the span is only a plane — and lies off it. Same recipe in any dimension: set up the linear combination, augment, row-reduce, and check for a contradiction row. A consistent system would instead hand you the weights c .

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.