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 ExampleDefinition: 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 dimen￾sions. 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