Definition If S is a nonempty subset of a vector space v, and S satisfying the following conditions (1).ax∈S, whenever x∈ S for ant scalar a. (2).x+y ES whenever x E S and y E S, then S is said to be a subspace of V Example S=Spal 2 is a one-dimensional subspace in R2 since/1 2|∈sand 2 (a+B) ∈S. 2 Therefore, the space spanned by a set of vectors in R has at most k dimen- ons. If this space has fewer than k dimensions, it is subspace, or hyperplane But the important point in the preceding discussion is that every set of vectors spans some space; it may be the entire space in which the vector reside, or some subspace of it. xercise LetS=f(r1, 12, I3)c1=x2. Show that S is a subspace of R 3.5 Rank of a matrix If A is an m x n matrix, each row of A is an n-tuple of real numbers and hence can be considered as a vector in RIxn. The m vectors corresponding to the rows of a will be referred to as the row vectors of A. Similarly, each column of A can be considered as a vector in rm and one can associate n column vectors with the matrix A Definition If A is an m x n matrix, the subspace of R xn spanned by the row vectors of A is called the row space of A. The subspace of Rm spanned by the column vectors of A is called the column space Example
Definition: If S is a nonempty subset of a vector space V, and S satisfying the following conditions: (1). αx ∈ S, whenever x ∈ S for ant scalar α. (2). x + y ∈ S whenever x ∈ S and y ∈ S, then S is said to be a subspace of V . Example: S= Span 1 2 is a one-dimensional subspace in R 2 since α 1 2 ∈ S and α 1 2 + β 1 2 = (α + β) 1 2 ∈ S. Therefore, the space spanned by a set of vectors in R k has at most k dimensions. If this space has fewer than k dimensions, it is subspace, or hyperplane. But the important point in the preceding discussion is that every set of vectors spans some space; it may be the entire space in which the vector reside, or some subspace of it. Exercise: Let S = {(x1, x2, x3) ′ |x1 = x2}. Show that S is a subspace of R 3 . 3.5 Rank of a Matrix If A is an m × n matrix, each row of A is an n−tuple of real numbers and hence can be considered as a vector in R 1×n . The m vectors corresponding to the rows of A will be referred to as the row vectors of A. Similarly, each column of A can be considered as a vector in R m and one can associate n column vectors with the matrix A. Definition: If A is an m × n matrix, the subspace of R 1×n spanned by the row vectors of A is called the row space of A. The subspace of R m spanned by the column vectors of A is called the column space. Example: 11
100 010 The row space of A is the set of all 3-tuples of the form a(100)+B(010)=(aB0) The column space of A is the set of all vectors of the form 0 0 1+0 Thus the row space of A is a two-dimensional subspace of Rx3 and the column space of A is R Theorem The column space and the row space of a matrix have the same dimension Definition The column(row) rank of a matrix is the dimension of the vector space that is spanned by its columns(rows) In short from this definition we know that the column rank is the number of linearly independent column of a matrix Theorem The column rank and row rank of a matrix are equal, that is rank(A)=rank(A')< min(number of rows, numbers of columns) Definition A full(short)rank matrix is a matrix whose rank is equal(fewer)to the number of columns it contains ome useful result rank(AB)< min(rank(A), rank(B))
Let A = 1 0 0 0 1 0 . The row space of A is the set of all 3-tuples of the form α(1 0 0) + β(0 1 0) = (α β 0). The column space of A is the set of all vectors of the form α 1 0 + β 0 1 + γ 0 0 . Thus the row space of A is a two-dimensional subspace of R 1×3 and the column space of A is R 2 . Theorem: The column space and the row space of a matrix have the same dimension. Definition: The column(row) rank of a matrix is the dimension of the vector space that is spanned by its columns (rows). In short from this definition we know that the column rank is the number of linearly independent column of a matrix. Theorem: The column rank and row rank of a matrix are equal, that is rank(A)=rank(A’)≤ min(number of rows, numbers of columns). Definition: A full (short) rank matrix is a matrix whose rank is equal (fewer) to the number of columns it contains. Some useful results: 1. rank(AB)≤ min(rank(A),rank(B)). 12