Key Concepts for Linear Algebra Test Preparation

Subspace

Check if the subset is closed under vector addition and scalar multiplication, and if it contains the zero vector.

Spanning Set

Check if the set of vectors can generate the entire vector space through linear combinations.

Linear Independence

Check if the only solution to the homogeneous equation is the trivial solution (all scalars zero), or row reduce the matrix and look for pivot positions.

Basis from Large Set

Remove vectors that are linear combinations of others until the set is linearly independent.

Basis from Small Set

Add vectors from the space that are linearly independent from the current set until the dimension is met.

Bases of the Same Space

They must contain the same number of vectors, equal to the dimension of the space.

Row Space/Column Space

Row space is the span of the row vectors; column space is the span of the column vectors. The rank is the dimension of either space.

Rank-Nullity Theorem

It relates the number of columns in a matrix to its rank and nullity: Rank + Nullity = Number of Columns.

Null Space

Empty null space means the matrix is injective (no free variables). Non-empty means the matrix has free variables (dependent columns).

Coordinate Vector

It lets you express a vector in terms of a basis, enabling transitions between coordinate systems.

Transition Matrix

Set up a matrix whose columns are the old basis vectors expressed in the new basis, then invert if necessary. It allows you to convert coordinates from one basis to another.

Inner Product

An inner product is a generalization of the dot product, satisfying linearity, symmetry, and positive-definiteness. Example: dot product, integral of product of functions.

Orthogonal Vectors

A set of vectors where each pair has a dot product of zero.

Angle Between Vectors

Use the cosine formula: cos(θ) = (u·v)/(||u||·||v||). Avoid it if vectors are orthogonal (angle is 90°) or parallel (angle is 0° or 180°).

Parallel Vectors

Vectors are parallel if one is a scalar multiple of the other. Distance is ||u − v||, length is ||v||.

Orthonormal Vectors

Vectors that are both orthogonal to each other and each have unit length.

Orthonormal Basis

Use the Gram-Schmidt process followed by normalizing each vector.

Gram-Schmidt Process

It turns a basis into an orthogonal set. Orthogonal projection removes components in certain directions. Normalizing sets vectors to unit length.