Linear Algebra – Vector Spaces, Row Spaces, Column Spaces (Lecture 09)

Definition

A vector space is a collection of objects (called vectors) where two operations are always valid:

Vector addition
Scalar multiplication

These operations must follow specific rules (axioms) so the system remains consistent.

A row space is the set of all linear combinations of the row vectors of a matrix.

A column space is the set of all linear combinations of the column vectors of a matrix.

Key Points

1. Vector Spaces

A vector space must satisfy 8 rules:

Vector Addition Rules

Commutative: ( $u + v = v + u$ )
Associative: ( $(u + v) + w = u + (v + w)$ )
Identity element: ( $u + 0 = u$ )
Inverse element: ( $u + (-u) = 0$ )

Scalar Multiplication Rules

Distributive: ( $(c + d)u = cu + du$ )
Associativity: ( $c(du) = (cd)u$ )
Identity scalar: ( $1u = u$ )

2. Sets and Spaces

A set is a collection of elements.
Empty set: ( $\emptyset = \{ \}$ )
A vector space is a set with defined rules for addition and scalar multiplication.

3. Vector Addition Example

In ( $\mathbb{R}^2$ ):

\begin{bmatrix} 1 \\ 2 \end{bmatrix} + \begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 3 \end{bmatrix}

Scaling:

3 \cdot \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \end{bmatrix}

4. Linear Combination

A vector ( w ) is a linear combination of vectors ( v_1, v_2, …, v_r ) if:

w = k_1 v_1 + k_2 v_2 + \cdots + k_r v_r

Where:

( $k_1, k_2, ..., k_r$ ) are scalars (coefficients)

5. Span

The span of a vector is all possible vectors created by scaling it.

Example:

x = \begin{bmatrix} 1 \\ 1 \end{bmatrix}

Then:

\text{span}(x) = { \alpha x : \alpha \in \mathbb{R} }

This creates a line through the origin.

Examples of span points:

$(1,1), (2,2), (0.5,0.5), (-1,-1)$

6. Span of Multiple Vectors

If we have:

v_1, v_2, ..., v_n

Then:

\text{span}(v_1, v_2, ..., v_n)

is all linear combinations:

a v_1 + b v_2 + c v_3 + ...

7. Basis

A basis is a set of vectors that:

Generate the space (span it)
Are linearly independent

Example in ( \mathbb{R}^2 ):

\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}

These generate the whole 2D plane.

8. Row Space

The row space is formed by all linear combinations of row vectors.
Row operations help determine:
- Rank
- Linear independence
- Dependence among equations

9. Column Space

The column space is formed by all linear combinations of column vectors.
It determines:
- Whether a system has solutions
- Whether a vector lies in the span

10. Solving Systems and Span

To check if a vector is in a span:

Convert to a system of linear equations
Solve using Gaussian elimination

If:

Solution exists → vector is in the span
No solution → not in span

Example / Code

Column Space Interpretation

Given system:

a v_1 + b v_2 + c v_3 = u

This becomes:

Ax = b

Solving determines if ( b ) is in the column space of A.

Output (if any)

If system is consistent → solution exists → vector is in column space
If inconsistent → no solution → vector is not in column space

Common Mistakes

1. Confusing span and basis

Span = all possible combinations
Basis = smallest independent generating set

2. Thinking vectors must start at origin incorrectly

Span always passes through origin

3. Mixing row and column space

Row space → rows of matrix
Column space → columns of matrix

4. Incorrect linear combination setup

Wrong:

av_1 + bv_2 = u \text{ without solving system}

Correct:

Convert into equations and solve.

Short Exam Notes (very concise revision points)

Vector space → closed under addition & scalar multiplication
8 axioms define vector space
Linear combination: ( w = \sum k_i v_i )
Span = all linear combinations
Basis = independent spanning set
Row space → span of rows
Column space → span of columns
Solve systems → check span membership
Gaussian elimination = key tool