Linear Algebra Exam 1

System of linear equations

A collection of one or more linear equations involving the same set of variables.

Consistent system

A linear system with at least one solution.

Inconsistent system

A linear system with no solution.

Elementary row operations

Replace a row by itself plus a multiple of another row; swap two rows; scale a row by a nonzero constant.

Row echelon form

Nonzero rows above all-zero rows, each leading entry right of the one above, entries below a leading entry are zero.

Reduced row echelon form (RREF)

Every leading entry is 1, and it is the only nonzero entry in its column.

Pivot column

A column containing a pivot position (a leading 1 in RREF).

Free variable

A variable not tied to a pivot column; can take any value.

Vector equation

x1a1 + x2a2 + ... + xnan = b, equivalent to a linear system with those vectors as columns.

Span of a set of vectors

The set of all linear combinations of those vectors.

Matrix equation Ax=b

Compact form of a linear combination of columns of A, weighted by entries of x.

When is Ax=b is consistent for EVERY b?

If and only if the columns of A span R^m (a pivot in every ROW of A).

When Ax=b has a UNIQUE solution

If and only if there are no free variables (a pivot in every COLUMN of A).

Homogeneous system Ax=0

Always has at least the trivial solution x=0.

When Ax=0 has nontrivial solutions

If and only if the system has at least one free variable.

Parametric vector form

x = p + t1v1 + t2v2 + ..., where p is one particular solution and vi span the homogeneous solutions.

Linearly independent vectors

The only solution to c1v1+...+cnvn=0 is all ci=0 (pivot in every column when set as a matrix).

Linearly dependent vectors

Some nontrivial combination of the vectors equals 0; at least one vector is a combo of the others.

Vectors that are automatically dependent

Any set containing the zero vector, OR any set with more vectors than entries per vector (more vectors than rows).

Linear transformation

T(u+v)=T(u)+T(v) and T(cu)=cT(u) for all vectors u,v and scalars c.

Standard matrix of T

Built by computing T(e1), T(e2), ..., T(en) and using the results as columns.

One-to-one transformation

Each output has at most one input; equivalent to Ax=0 having ONLY the trivial solution (columns independent).

Onto transformation

Every vector in the codomain is hit by some input; equivalent to columns of A spanning the codomain (pivot in every row).