Linear algebra - Chapter 1

What is a consistent system?

A system of equations that has at least one solution.

What is an inconsistent system?

A system of equations that has no solution.

What is an echelon form (row echelon form)?

A matrix where: all nonzero rows are above zero rows, each leading entry is to the right of the one above, and all entries below a leading entry are zero.

What is reduced echelon form (RREF)?

Echelon form where each leading entry is 1 and the leading 1 is the only nonzero entry in its column.

What is a pivot position?

The first non-zero position in a row.

What is a pivot column?

A column that contains a pivot position.

When is a vector b in the span of vectors v₁,…,vₙ?

When the system Ax = b is consistent.

What does it mean for Ax = b to have a solution?

It means b is a linear combination of the columns of A.

What four statements are equivalent to Ax = b being consistent?

1) Ax = b has at least one solution; 2) b is a linear combination of A’s columns; 3) b is in the column space of A; 4) The augmented matrix [A b] is consistent.

What is a homogeneous system?

A system of the form Ax = 0.

Is a homogeneous system always consistent?

Yes, because x = 0 is always a solution.

What is a trivial solution?

The solution x = 0 to the equation Ax = 0.

What is a nontrivial solution?

A nonzero solution to Ax = 0.

What is the structure of the solution set of Ax = b?

(All solutions) = (one particular solution) + (the general solution of Ax = 0).

What is linear independence?

A set of vectors is linearly independent if the equation x₁v₁ + … + xₙvₙ = 0 has only the trivial solution.

What is linear dependence?

A set of vectors is linearly dependent if there is a nontrivial solution to x₁v₁ + … + xₙvₙ = 0.

What does it mean if one vector in a set is a linear combination of the others?

The set is linearly dependent.

What does it mean that T(x) = Ax?

Every linear transformation from ℝⁿ to ℝᵐ is matrix multiplication by some matrix A.

When is a linear transformation T: ℝⁿ → ℝᵐ onto?

When A has a pivot in every row, meaning its columns span ℝᵐ.

When is a linear transformation T: ℝⁿ → ℝᵐ one-to-one?

When A has a pivot in every column, meaning its columns are linearly independent.

What does it mean for Ax = 0 to have only the trivial solution?

It means the columns of A are linearly independent.

What does it mean for Ax = 0 to have infinitely many solutions?

It means the columns of A are linearly dependent.

What is a free variable vs free variable ?

A free variable in a system corresponding to a non-pivot column.
A basic variable corresponds to a pivot column.

What is the general homogeneous solution?

A linear combination of special solution vectors corresponding to free variables.

Why does a pivot in every column imply one-to-one (uniqueness)?

Because it forces Ax = 0 to have only the trivial solution.

Why does a pivot in every row imply onto?

Because the columns span the entire codomain.

What is the difference between a particular solution and a homogeneous solution?

A particular solution solves Ax = b; a homogeneous solution solves Ax = 0.

What does it mean for a transformation to be unqiue?

It means that no two different input vectors map to the same output vector

What are the conditions for a matrix to be unique?

For a matrix to be unique it means that the matrix has to be square and the identity matrix.

What characteristic do linear transformations have?

• Linear transformations preserve the operations of vector addition and scalar multiplication

• T(0) = 0

• T(cu + dV) = cT(u) + dT(V)

Provide the standard matrix for shear

\left\lbrack\frac10\cdot\frac{k}{1}\right\rbrack