Linear Algebra Lecture 11
Linear Algebra Lecture 11
1. Matrix space
2. Bases of new vector spaces
3. Rank one matrices
Matrix space
Each 3 \times 3 matrix acts as a vector within the vector space. These matrices function as vectors and are capable of being added together. Additionally, they can be multiplied by scalar quantities to produce new vectors. A combination of these matrices can be formed to create more complex structures within the vector space.
Subspace of the matrix space
Each symmetric matrix is characterized by its symmetry.
The set of all upper-triangular matrices forms a specific class.
The collection of all diagonal matrices represents another important category.
Basis for M = all matrices
The standard basis for our matrix space consists of the set of elementary matrices, each having exactly one entry equal to 1 and all other entries equal to 0. Specifically, these matrices are:
\begin{bmatrix}1&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix}, \begin{bmatrix}0&1&0\\ 0&0&0\\ 0&0&0\end{bmatrix}, \begin{bmatrix}0&0&1\\ 0&0&0\\ 0&0&0\end{bmatrix}, \ldots, \begin{bmatrix} ... \end{bmatrix}.
Our space is isomorphic to a nine-dimensional vector space, with the distinction that the components are arranged in a square format rather than a column vector structure.
Dimension and basis for subspace
| Subspace | Basis | Dimension |
|---|---|---|
| Original | \begin{bmatrix} 1&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}, \begin{bmatrix} 0&1&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}, ..., \begin{bmatrix} 0&0&0\\ 0&0&0\\ 0&0&1 \end{bmatrix} | 9 |
| Symmetric | \begin{bmatrix} 1&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}, \begin{bmatrix} 0&0&0\\ 0&1&0\\ 0&0&0 \end{bmatrix}, \begin{bmatrix} 0&0&0\\ 0&0&0\\ 0&0&1 \end{bmatrix},\begin{bmatrix} 0&1&0\\ 1&0&0\\ 0&0&0 \end{bmatrix},\begin{bmatrix} 0&0&0\\ 0&0&1\\ 0&1&0 \end{bmatrix}, \begin{bmatrix} 0&0&1\\ 0&0&0\\ 1&0&0 \end{bmatrix} | 6 |
| Upper triangular | \begin{bmatrix} 1&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}, \begin{bmatrix} 0&0&0\\ 0&1&0\\ 0&0&0 \end{bmatrix}, \begin{bmatrix} 0&0&0\\ 0&0&0\\ 0&0&1 \end{bmatrix},\begin{bmatrix} 0&1&0\\ 0&0&0\\ 0&0&0 \end{bmatrix},\begin{bmatrix} 0&0&0\\ 0&0&1\\ 0&0&0 \end{bmatrix}, \begin{bmatrix} 0&0&1\\ 0&0&0\\ 0&0&0 \end{bmatrix} | 6 |
| Diagonal | \begin{bmatrix} 1&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}, \begin{bmatrix} 0&0&0\\ 0&1&0\\ 0&0&0 \end{bmatrix}, \begin{bmatrix} 0&0&0\\ 0&0&0\\ 0&0&1 \end{bmatrix} | 3 |
Intersection : S \cap U = symmetric and upper triangular = diagonal
dim(S \cap U) = 3
Sum : S + U = any element of S + any element of U = all matrices
dim(S + U) = 9
dim(S) + dim(U) = dim(S \cap U) + dim(S + U)
Solution spaces
A vector space does not contain vectors within it; instead, it originates from differential equations.
All solutions of \frac{d^2y}{dx^2} + y = 0
All combinations y = c_1cosx + c_2sinx
dimension and basis
base : cosx and sinx; the dimension of the solution space itself equals two since we have a second-order equation.
Rank one matrices
Example 1: A = \begin{bmatrix} 1 &4&5 \\ 2&8&10 \end{bmatrix}
The dimension of column space and row space of A is one.
dim C(A)= rank = dim C(A^T) = 1
A = \begin{bmatrix} 1 &4&5 \\ 2&8&10 \end{bmatrix} 可以表示为 A = \begin{bmatrix} 1 \ 2\end{bmatrix}\times \begin{bmatrix} 1 &4&5\end{bmatrix}$
Every rank one matrix has the form one column times one row. A = u v^T
Rank one matrices are like building blocks, can produce every matrix.
集合中秩为一的矩阵并不是一个子空间
Example 2:
Within the four-dimensional real space R^4 , a vector v is represented as [v_1, v_2, v_3, v_4]^T , where each component of v is denoted by v_i . The set S comprises all such vectors v for which their components satisfy the equation v_1 + v_2 + v_3 + v_4 = 0 . Additionally, it can be shown that S corresponds to the nullspace of the matrix A = [1\ 1\ 1\ 1]^\top .
Rank of A = r = 1
dim N(A) = n - r = 4 - 1 = 3
The basis is defined by assigning the values at indices 0 and 1 to the free variables.
C(A) = R^1 , dim C(A) = 1
N(A^T) = \{ 0 \}
该矩阵的零空间维数为三，
其行空间维数为一，
列空间维数为一，
左零空间维数为零。
Among all subspaces, the minimal one constitutes an empty set, which contains precisely a single element. This unique element exhibits a zero-dimensional nature, and its basis remains unformed.