MATH 3320: Linear Algebra Ch 1 Review

The three elementary row operations are:

Scaling, interchange, and replacement

Scaling:

Multiply all entries in a row by a nonzero constant

Interchange:

Interchange two rows

Replacement:

Replace one row by the sum of itself and a multiple of another row

Two matrices are row equivalent if:

There is a sequence of elementary row operations that transforms one matrix into another

A system of linear equations is said to be consistent if:

It has either one solution or infinitely many solutions

A system of linear equations is said to be inconsistent if:

It has no solution

A pivot column is:

A column that contains a pivot

A pivot is:

the leading entry in a row of a matrix that’s in echelon form

A free variable is:

A term for a parameter used in a system with infinitely many solutions.

Span { v1, · · · , vp} is:

the collection of all vectors that can be written in the form c1v1 + c2v2 + … + cpvp; with c1, …, cp any real numbers

An indexed set of vectors {v1, … , vp} in Rn is said to be linearly independent if:

the vector equation x1v1 + x2v2 + …. + xpvp = 0 has only the trivial solution

An indexed set of vectors {v1, · · · , vp} in Rn is said to be linearly dependent if

the vector equation x1v1 + x2v2 + …. + xpvp = 0 has nontrivial solution

A transformation T is linear if

T(u + v) = Tu + Tv and T(cu) = cT(u)

A transformation T : Rn −→ Rm is onto Rm if

for each vector b in Rm, there is at least one x in Rn such that T(x) = b.

A transformation T : Rn −→ Rm is one-to-one if:

Iff the equation T(x) has only the trivial solution.

Thm 1

Each matrix is row equivalent to one and only one reduced echelon matrix

Reduced Row Echelon Form

When the pivot in each nonzero row is 1 and is the only nonzero entry in its column.

Thm 2:

A linear system is consistent iff the rightmost column of the RREF of the augmented matrix is not a pivot column.,

Thm 5:

If A is a mxn matrix, u and v are vectors in Rn, and c is a scalar, then A(u +v) = Au + Av and A(cu) = cA(u)

Thm 4: Let A be a mxn matrix. Then the following statements are equivalent

1. For each b in Rm, the equation Ax=b has a solution

2. Each b in Rm is a linear combination of the columns a1, …, an of A

3. Each b in Rm is generated by the columns of a1, …, an of A

4. Span{a1, …., an} = Rm (ie. the columns of A span Rm)

5. A has a pivot in every row

Thm 7:

An indexed set S = {v1, …, vp} of two or more vectors is linearly dependent iff at least one of the vectors in S is a linear combination of the others.

Thm 10: Let T: Rn —> Rm be a linear transformation.

Then there exists a unique matrix A such that T(x) = Ax for all x in Rn

Thm 11: Let T: Rn —> Rm be a linear transformation.

T is one-to-one iff the equation T(x) = 0 has only the trivial solution

Thm 12: Let T: Rn —> Rm be a linear transformation and let A be the standard matrix for T (1-1).

T is one-to-one iff the columns of A are linearly independent.

Trick to recognize onto:

No rows of all zeros, every row has a pivot, m >= n.

Tricks to recognize one-to-one

Pivot in every column, trivial solution, n >= m