1. Ch 6 Vector Spaces

2. Vector Space Axioms X,Y,Z elements of  and α, β elements of  Def of vector addition Def of multiplication of scalar and vector These defs satisfy the 10 axioms (pg. 155) Addition –Uniqueness, closure –Commutativity –Associativity –Identity –Inverse Scalar Multiplication –Uniqueness, closure –Associativity –Right Distributivity –Left Distributivity –Unit scalar mult.

3. Subspaces A set  is a subspace of a vector space  if –Every element of  is in , and –  is a vector space. A line through the origin is a subspace of 2-D Euclidean space A plane that contains the origin is a subspace of 3-D Euclidean space

4. Linear Independence of Vectors The vectors A 1, A 2, … A n over a fieled  are linearly independent if –any set {k 1, k2, … k n } of elements of , for which At least one k i is zero.

5. Linear Combinations Consider a set of vectors, and a set of scalars A linear combination of the vectors is or

6. Linear Dependence and Rank Multiplication of a matrix by a column vector produces a linear combination of the columns Multiplication of a matrix by a matrix can be viewed as multiplication of left matrix by each column of right matrix. Full column rank implies columns are linearly independent Rank deficient matrices yield infinite solutions

7. Range or Image View a matrix as a collection of column vectors. Consider the set of vectors formed by an arbitrary linear combination of the column vectors (result of multiplying a matrix by an arbitrary column vector) The range space or the image of a matrix is the set of vectors generated by multiplying the matrix by an arbitrary column vector.  (A)=

8. Basis If the columns of A are linearly independent, they are called a basis for  (A) If A has full column rank, n, and vector X in  (A) is a unique linear combination of the basis vectors in A, i.e. X = AK has a unique solution for K. The entries in K are the coordinates of X wrt the Basis A.

9. Dimension The dimension of a vector space is the number of (linearly independent) vectors in a basis for the space.

10. Standard Basis The column vectors in the identity matrix for the standard basis [e 1 e 2 …e n ]=I n Remember i, j, k from physics AX=B has a solution only if each and every column of B is in  (A), i.e. –  (B) is a subset of  (A) This can be tested by –constructing the matrix [A B]; –computing an upper-row compression (Sec. 5.7) and noting that  (B) is a subset of  (A) if and only if P 2 B=0

11. Null Space or Kernel The null space, or kernel, of a matrix is the set  (A) = {X: AX = 0} Set of solutions of the homogeneous equation AX = 0 –Contains non zero vectors only if rank(A) < cols(A) If X is a solution of AX=B, then so is X’=X+H for any H in  (A) –Hence solution is unique only if A has full column rang

12. Basis for  (A) and  (A)

13. Orthogonal Basis

14. Change of Basis

15. Similarity Transformation