1.1 Vectors and matrices
Linear Combination: A linear combination of a set of vectors is a sum of those vectors where each vector is multiplied by a scalar coefficient.
Subspace: A subspace of a vector space is a subset of the space that is itself a vector space under the same addition and scalar multiplication operations. Specifically, a subspace must be closed under vector addition and scalar multiplication.
※ From the definition, we can see that a subspace must contain the zero vector.
Basis: A basis of a vector space is a set of linearly independent vectors that span the entire space. In other words, every vector in the space can be expressed as a unique linear combination of the basis vectors.
※ Although a vector space has many different bases, the number of vectors in the basis is the same.
Standard Basis: The standard basis of a vector space consists of vectors where each vector has a 1 in one coordinate and 0 in all other coordinates. By performing elementary transformations on the matrix formed by the basis, we can obtain the standard basis.
Dimension: The dimension of a vector space is the number of vectors in any basis of the space.
Column Space: The column space of a matrix is the set of all possible linear combinations of its column vectors. It is a subspace of the vector space that the columns are part of. In other words, it represents all the vectors that can be formed by taking linear combinations of the columns of the matrix.
※ The dimension of the column space is not equal to the dimension of the vector in the column space
1.2 Rank of a matrix
Rank: The rank of a matrix is the dimension of its column (or row) space. It represents the maximum number of linearly independent columns (or rows) in the matrix. The rank provides information about the matrix’s ability to span its column or row space.
In practical terms, the rank tells you how many dimensions of the space can be reached or represented by the matrix. For example, if a matrix has rank 3, it means that there are 3 linearly independent columns (or rows) and that the column space (or row space) of the matrix is 3-dimensional.
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