Linear Algebra: Vector Spaces, Subspaces, and Matrix Properties

Vocabulary and key concepts regarding vector spaces, subspaces, matrix dimensions, and bases for null, column, and row spaces.

Vector space

A set closed under vector addition and scalar multiplication.

Subspace

A subset of a vector space that contains the zero vector, is closed under addition, and is closed under scalar multiplication.

Subspace tests

1. Contains the zero vector; 2. Closed under addition; 3. Closed under scalar multiplication.

Null space ($Nul(A)$, Kernel)

$Nul(A) = \{x : Ax = 0\}$. It is a subspace of $\mathbb{R}^n$ (the number of columns).

Column space ($Col(A)$, Range/Image)

The span of the columns of $A$. It is a subspace of $\mathbb{R}^m$ (the number of rows).

Row space ($Row(A)$, Row Set)

The span of the rows of $A$. It is a subspace of $\mathbb{R}^n$ (the number of columns).

Span

All linear combinations of a set of vectors.

Basis

A linearly independent set that spans a vector space.

Linearly independent

A condition where the only solution to $c_1v_1 + \dots + c_nv_n = 0$ is the trivial solution.

Linearly dependent

A condition where at least one vector is a linear combination of the others.

Dimension

The number of vectors in any basis of a vector space.

Rank

The number of pivot columns in a matrix, which determines the dimension of $Col(A)$ and $Row(A)$..

Nullity

The number of free variables, which determines the dimension of $Nul(A)$..

Rank-Nullity Theorem

$\text{Rank} + \text{Nullity} = \text{Number of Columns}$.

Standard basis of $P_2$

$$\{1, t, t^2\}$$

Standard basis of $P_3$

$$\{1, t, t^2, t^3\}$$

Pivot variable

A variable corresponding to a pivot column.

Free variable

A variable corresponding to a non-pivot column, identified by columns without pivots.

Redundant vector

A vector that is a linear combination of the others and should be removed from a spanning set to obtain a basis.

Basis for $Nul(A)$ (Procedure)

Solve $Ax = 0$ and use the special solution vectors.

Basis for $Col(A)$ (Procedure)

Find pivot columns in the RREF, then take those corresponding columns from the ORIGINAL matrix.

Basis for $Row(A)$ (Procedure)

Use the nonzero rows of the RREF.

Maximum rank of an $m \times n$ matrix

$$\min(m, n)$$

Maximum nullity of an $m \times n$ matrix

$n$ (the number of columns).

Minimum nullity of an $m \times n$ matrix

$$n - \min(m, n)$$