Vectors, Matrices and Systems
Cheat sheet.

The key rules, formulas and reminders from all 3 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Vectors, Linear Combinations and Systems

A\mathbf x=x_1\mathbf a_1+\cdots+x_n\mathbf a_n.

Match input length to the number of columns. A system is solvable exactly when its target belongs to the span of the columns.

Row-reduce the full augmented matrix. Pivot columns identify dependent variables; nonpivot variable columns are free.

A pivot in the augmented column signals inconsistency. Count free variables only after checking consistency.

$(AB)\mathbf x=A(B\mathbf x)$ . Solve $LU\mathbf x=\mathbf b$ by forward then backward substitution; with swaps use $PA=LU$ .

Multiplication is generally not commutative. Not every square matrix has an inverse.

2 reference blocks

A\mathbf x=x_1\mathbf a_1+\cdots+x_n\mathbf a_n.

Match input length to the number of columns. A system is solvable exactly when its target belongs to the span of the columns.

2 reference blocks

Row-reduce the full augmented matrix. Pivot columns identify dependent variables; nonpivot variable columns are free.

A pivot in the augmented column signals inconsistency. Count free variables only after checking consistency.

2 reference blocks

$(AB)\mathbf x=A(B\mathbf x)$ . Solve $LU\mathbf x=\mathbf b$ by forward then backward substitution; with swaps use $PA=LU$ .

Multiplication is generally not commutative. Not every square matrix has an inverse.