Matrices Lecture 3

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Questions and Answers

What is the dimension of the vector space of linear maps from ℝ² to ℝ³?

9 (correct)

What is the relationship between the matrix representing a linear map and the linear map itself?

The matrix represents the linear map with respect to standard bases (correct)

The linear map represents the matrix with respect to standard bases

The linear map is an isomorphism of the matrix

The matrix is an isomorphism of the linear map

What is the condition for a matrix to be invertible?

The column vectors of the matrix are linearly dependent

The row vectors of the matrix are linearly dependent

The row vectors of the matrix are linearly independent

The column vectors of the matrix are linearly independent (correct)

What is the relationship between the rank of a matrix and the rank of its transpose?

The rank of a matrix is equal to the rank of its transpose Signup and view all the answers

What is the definition of the rank of a matrix?

The rank of the linear map represented by the matrix Signup and view all the answers

What is the purpose of a change-of-coordinates matrix?

To change the coordinates of a vector from one basis to another Signup and view all the answers

What is the result of composing two linear maps?

A linear map from the domain of the first map to the codomain of the second map Signup and view all the answers

What is the relationship between the vector space of matrices and the vector space of linear maps?

The vector space of matrices is isomorphic to the vector space of linear maps Signup and view all the answers

What is the matrix representing the linear map 𝑓 with respect to the bases 𝐵 and 𝐵′ of 𝑉 and 𝑊 respectively?

ℳ(𝑓, 𝐵, 𝐵′) = [𝑎11 𝑎12 ⋯ 𝑎1𝑛; 𝑎21 𝑎22 ⋯ 𝑎2𝑛; ⋮ ⋮ ⋮ ⋮ ; 𝑎𝑚1 𝑎𝑚2 ⋯ 𝑎𝑚𝑛] Signup and view all the answers

What is the matrix representing the identity map 𝐼𝑑𝑉 with respect to a basis 𝐵 of 𝑉?

The identity matrix 𝐼𝑛 Signup and view all the answers

What is the matrix representing the linear map 𝑔 ∘ 𝑓: 𝑈 → 𝑊 with respect to 𝐵1 and 𝐵3?

The matrix product 𝐵𝐴 Signup and view all the answers

What is the basis of 𝑉 in Example 18?

𝐵 = 𝑒1, 𝑒2 Signup and view all the answers

What is the definition of a matrix in this lecture?

An 𝒎 × 𝒏 array of scalars from a given field 𝐾 Signup and view all the answers

What is the result of applying the linear map 𝑓 to the vector (1,0) in Example 18?

(1,2,1) Signup and view all the answers

What is the size of a row matrix of size 1 × 𝑛?

1 × 𝑛 Signup and view all the answers

What is the result of applying the linear map 𝑓 to the vector (0,1) in Example 18?

(−1,1,3) Signup and view all the answers

What is the matrix representing the linear map 𝑓 with respect to the standard bases 𝐵 and 𝐵′?

[1 −1; 2 1; 1 3] Signup and view all the answers

What is the size of a column matrix of size 𝑚 × 1?

𝑚 × 1 Signup and view all the answers

What are the bases of 𝑉 and 𝑊 in Definition 7?

𝐵 = 𝑣1, 𝑣2, ⋯, 𝑣𝑛 and 𝐵′ = 𝑤1, 𝑤2, ⋯, 𝑤𝑚 Signup and view all the answers

What is the notation for a matrix 𝐴 of size 𝑚 × 𝑛?

𝐴 = 𝑎𝑖𝑗, 1≤𝑖≤𝑚, 1≤𝑗≤𝑛 Signup and view all the answers

What is the condition for a matrix 𝐴 to be diagonal?

𝑎𝑖𝑗 = 0 for 𝑖 ≠ 𝑗 Signup and view all the answers

What is the definition of a square matrix?

A matrix 𝐴 is said to be square if 𝑚 = 𝑛 Signup and view all the answers

What is the field 𝐾 in the context of matrices?

An arbitrary field Signup and view all the answers

What is the notation for a matrix 𝐴 in the context of matrices?

𝐴 = 𝑎𝑖𝑗, 1≤𝑖≤𝑚, 1≤𝑗≤𝑛 Signup and view all the answers

Study Notes

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.

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Description

This lecture covers the basics of matrices, including definitions, operations, vector space, ring of square matrices, invertible matrices, and more.