Systems, Matrices, and Row Reduction
Core vocabulary and structural ideas behind Gaussian elimination, pivots, and solution sets.
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Questions Covered in This Set
12 cards to master
What is the 'row picture' of a linear system?
Each equation is a hyperplane (a plane in R^3); a solution is a point lying on all of them. They can meet in a point, a line, a plane, or nowhere.
What question does the 'column picture' ask?
Can b be written as a linear combination of the columns of A? Solutions exist exactly when b lies in the span of the columns.
What is the 'operator picture' of Ax = b?
A is a function R^n → R^m; we ask which inputs x the matrix sends to the output b.
List the three elementary row operations.
(1) Swap two rows, (2) scale a row by a nonzero constant, (3) add a multiple of one row to another.
Why do row operations preserve the solution set?
Each operation is reversible, so any solution of the new system solves the old one and vice versa. Scaling by 0 is banned because it isn't reversible.
A reduced row of [A|b] reads 0 0 0 | 1. What does that mean?
The system is inconsistent — no solutions — because b is not in the span of the columns of A.
Define reduced row echelon form (RREF).
Each nonzero row's leading entry is 1, that 1 is the only nonzero entry in its column, and the leading 1s move strictly right as you go down.
Why does the uniqueness of RREF matter?
Because the data you read off (pivots, rank, free variables) is intrinsic to A, not an artifact of the elimination choices you made.
Pivot column vs. free variable?
A column with a leading 1 is a pivot column and its variable is basic; every non-pivot column corresponds to a free variable you may set arbitrarily.
What is the general form of a solution set with free variables?
A particular solution plus a parameterized family of homogeneous solutions, e.g. (1,0,3) + y(−2,1,0) — a line through the particular point.
What is the rank of a matrix?
The number of pivots in its RREF; it measures how many independent directions the columns (or rows) contribute.
In the lecture's example, why did columns span only a plane?
The second column was exactly twice the first, so it added nothing new — only two independent columns in R^3.
