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Unit Three Review Linear Spaces Review of terms: A Linear Space: is a nonempty set that is closed under addition and multiplication by a scalar (real or complex) that satisfies under addition: commutative property, associative property, additive identity, and additive inverse property and under multiplication by a scalar (k1 k 2 )v1 k1v1 k 2 v1 , (k1k 2 )v1 k1 (k 2 v1 ) , k1 (v1 v2 ) k1v1 k1v2 and 1* v v . Subspace: a nonempty subset S of a linear space V that inherits the same properties as the space itself. In other words, a subspace must be closed under addition and under multiplication by a scalar. Linear Combination: A vector w is a linear combination of a set of vectors v1 , v2 ,..., vn if there exist scalars c1 , c2 ,..., cn such that w c1v1 c2 v2 ... cn vn . Span of a set L(S): is the set of all linear combination c1v1 c2 v2 ... cn vn of the elements v1 , v2 ,..., vn of the set S. Linear Dependent Set: set of vectors v1 , v2 ,..., vn such that there exists at least one nonzero scalar for which c1v1 c2 v2 ... cn vn 0 . Linear Independent Set: set of vectors v1 , v2 ,..., vn for which if c1v1 c2 v2 ... cn vn 0 , then c1 c2 c3 ... cn 0 . Basis of a Linear Space: is a set of linearly independent set that spans the space. Row Space of a Matrix: is the subspace spanned by the rows of the matrix. Column Space of a Matrix: is the subspace spanned by the columns of the matrix. Rank of a Matrix: is the maximum number of linear independent rows in the reduced row echelon form of the matrix. Review of some facts: The linear span of a set S is a subspace. The system Ax b is consistent if and only if rank (A | b) =rank (A). A basis of a linear space is not unique but the number of elements is the same for all bases. A vector in a linear space can be expressed uniquely in terms of the basis elements. Review Questions: 1. Give an example of a linear space and state the reasons for your choice. 2. Describe the subspaces of R 3 . 3. For what values of a, b, and c is the vector ( a, b, c ) a linear combination of the vectors: (1,2,4), (1,1,2) . Write one vector that can be written asa linear combination. 4. For what values of a, b, and c is the vector ( a, b, c ) a linear combination of the vectors: (1,2,4), (3,1,2) and (5,1,3) . 5. If an nxn matrix A has n independent rows, describe the solution set of the system Ax b . 6. If an nxn matrix A has r (r < n) independent rows, describe the solution set of the system Ax b . 7. For what values of x: 2 1 1 3 1 2 5 2 x does the matrix have rank 2? 8. Given the matrix A:: 2 1 1 1 3 4 5 5 6 (a) Are the rows of A linearly dependent or independent? Explain (b) Describe the row space and the column space of the matrix 9. Find a basis for the row space of the matrix A: 2 1 1 1 3 4 1 4 2