Matrices Lecture 3

Matrix Definitions

• A matrix is an 𝑚 × 𝑛 array of scalars from a given field 𝐾.
• The individual values 𝑎𝑖𝑗 in the matrix are called entries.
• The size of the array is 𝒎 × 𝒏, where 𝑚 is the number of rows and 𝑛 is the number of columns.

Row Matrix and Column Matrix

• A row matrix of size 1 × 𝑛 is a matrix of the form 𝑎11 ⋯ 𝑎1𝑛.
• A column matrix of size 𝑚 × 1 is a matrix of the form ⋮ 𝑎𝑚1.

Notation

• A matrix 𝐴 of size 𝑚 × 𝑛 is denoted by 𝐴 = 𝑎𝑖𝑗 1≤𝑖≤𝑚 or 𝑎𝑖𝑗 when we know the number of rows and columns.

Square Matrices

• If 𝑚 = 𝑛, the matrix is called square.
• A square matrix is said to be diagonal if 𝑎𝑖𝑗 = 0 for 𝑖 ≠ 𝑗.

Matrix of a Linear Map

• The matrix representing the linear map 𝑓 with respect to the bases 𝐵 and 𝐵′ of 𝑉 and 𝑊 respectively is ℳ(𝑓, 𝐵, 𝐵′).
• Each vector 𝑓(𝑣𝑖), 𝑖 ∈ 1,2, ⋯ , 𝑛, can be written with respect to the basis 𝑤1, 𝑤2, ⋯, 𝑤𝑚 of 𝑊 as 𝑓(𝑣𝑖) = 𝑎𝑖1 𝑤1 + 𝑎𝑖2 𝑤2 + ⋯ + 𝑎𝑖𝑚 𝑤𝑚.

Matrix of the Identity Map

• The matrix representing the identity map 𝐼𝑑𝑉 with respect to a basis 𝐵 of 𝑉 is the identity matrix 𝐼𝑛.

Composed Linear Maps Matrix

• Let 𝑓: 𝑈 → 𝑉 and 𝑔: 𝑉 → 𝑊 be two linear maps, and 𝐴 the matrix representing 𝑓 with respect to 𝐵1 and 𝐵2, and 𝐵 the matrix representing 𝑔 with respect to 𝐵2 and 𝐵3.
• Then 𝐵𝐴 is the matrix representing the linear map 𝑔 ∘ 𝑓: 𝑈 → 𝑊 with respect to 𝐵1 and 𝐵3.

Isomorphism between Vector Space of Matrices and Vector Space of Linear Maps

• There exists an isomorphism 𝜑 from the 𝐾-vector space ℒ(𝑉, 𝑊) of linear maps to ℳ𝑚,𝑛(𝐾) that sends a linear map 𝑓: 𝑉 → 𝑊 to the matrix representing 𝑓 with respect to 𝐵 and 𝐵′.

Some Properties

• A matrix 𝐴 ∈ ℳ𝑛(𝐾) is invertible if, and only if, the linear map representing 𝐴 is bijective.
• The matrix 𝐴 is invertible if, and only if, the column vectors of 𝐴 are linearly independent.

Matrix Rank

• The rank of a matrix 𝐴 ∈ ℳ𝑝,𝑞(𝐾) is the rank of the linear map representing 𝐴.
• It is denoted by 𝑟(𝐴).
• For any matrix 𝐴, we have 𝑟(𝐴) = 𝑟(𝐴𝑇).

Change-of-Coordinates Matrix

• Let 𝑉 be a 𝐾-vector space of dimension 𝑛.
• The change-of-coordinates matrix is used to transform between different bases of 𝑉.