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Unformatted text preview: Unit III: Subspaces, Bases, and Dimension 1. Subspaces of R n Definition: A subset V of R n is a subspace if (1) V is closed under addition, i.e., if x , y ∈ V , then x + y ∈ V , (2) V is closed under scalar multiplication, i.e., if x ∈ V and c ∈ R , then c x ∈ V , (3) ∈ V . The conditions (1) and (2) are equivalent to requiring that V is closed under taking linear combinations. The condition (3) simply guarantees that V is nonempty. We can rephrase the definition of vector space as follows: Alternate definition: A subspace of R n is a nonempty subset of R n that is closed under taking linear combinations. Examples: The set { } consisting of the vector by itself is a subspace of R n . The set R n of all vectors is a subspace of R n . These two subspaces are referred to as the trivial subspaces of R n . Lines through the origin and planes containing the origin are subspaces. On the other hand, for u 6 = , the ray through u consisting of all vectors t u , t ≥ 0 is not a subspace of R n . It is not closed under scalar multiplication by 1. Other examples of subspaces are given by the following lemmas. Lemma. Let T be a linear transformation from R n to R m . Then the null space of T is a subspace of R n , and the image of T is a subspace of R m . Proofsketch. Check (1), (2), and (3) above. Lemma. If A is an m × n matrix, then the solution space of the homogeneous system A x = is a subspace of R n , and the image of A is a subspace of R m . Proofsketch. Check (1), (2), and (3) above. Definition: The linear span of a set of vectors is the collection of all finite linear combi nations of vectors in the set. The vectors v 1 ,..., v k span V if every vector in V is a linear combination of the v j ’s. Lemma. The linear span of a (nonempty) subset of R n is a subspace of R n . Proofsketch. Check (1), (2), and (3) above. Definition: The column space of a matrix is the linear span of the column vectors of the matrix. Lemma. The column space of a matrix A coincides with the image of A , that is, the image of the associated linear transformation T ( x ) = A x . Proof. If a 1 ,..., a n are the column vectors of the matrix A , then the vectors in the image of A are the vectors of the form A x = ∑ x j a j for x ∈ R n , and these are precisely the linear combinations of the columns of A . Definition: The row space of a matrix is the linear span of the row vectors of the matrix. The row space of an m × n matrix is a subspace of R n , while the column space of a matrix is a subspace of R m . 1 The row space of a matrix is the column space of its transpose. Example: Let the matrix A with transpose A t be given by A = 0 1 1 1 0 , A t = 1 1 1 0 ....
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 Spring '08
 lee
 Linear Algebra, Addition, Multiplication, Vectors, Scalar, UJ, A. Proofsketch

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