Transposes, permutations, spaces Rn In this lecture we introduce vector spaces and their subspaces. Permutations Multiplication by a permutation matrix P swaps the rows of a matrix; when applying the method of elimination we use permutation matrices to move ze ros out of pivot positions. Our factorization A = LU then becomes PA = LU, where P is a permutation matrix which reorders any number of rows of A. Recall that P−1 = P T , i.e. that P T P = I. Transposes When we take the transpose of a matrix, its rows become columns and its columns become rows. If we denote �the �entry in row i column j of matrix A by Aij , then we can describe A T by: A T ij = A ji . For example: ⎤T � 1 3 ⎣ 2 3 ⎦ = 1 3 4 1 ⎡ 2 3 4 1 � . A matrix A is symmetric if A T = A. Given any matrix R (not necessarily � �T � �T square) the product R T R is always symmetric, because R T R = R T R T = � �T R T R. (Note that R T = R.) Vector spaces We can add vectors and multiply them by numbers, which means we can dis cuss linear combinations of vectors. These combinations follow the rules of a vector space. One such vector space is R2 , the set of �all vectors with exactly two real � a number components. We depict the vector by drawing an arrow from b the origin to the point ( a, b) which is a units to the right of the origin and b units above it, and we call R2 the “x − y plane”. Another example of a space is Rn , the set of (column) vectors with n real number components. Closure The collection of vectors with exactly two positive real valued components is not a vector space. The sum of any two vectors in that collection is again in the collection, but multiplying any vector by, say, −5, gives a vector that’s not in the collection. We say that this collection of positive vectors is closed under addition but not under multiplication. If a collection of vectors is closed under linear combinations (i.e. under addition and multiplication by any real numbers), and if multiplication and addition behave in a reasonable way, then we call that collection a vector space. Subspaces A vector space that is contained inside of another vector space is called a sub space of that space. For example, take any non-zero vector v in R2 . Then the 2 set of all vectors cv, where c is a real number, � forms � a subspace of R . This 0 collection of vectors describes a line through in R2 and is closed under 0 addition. A line in R2 that does not pass through the origin is not a subspace of R2 . Multiplying any vector on that line by 0 gives the zero vector, which does not lie on the line. Every subspace must contain the zero vector because vector spaces are closed under multiplication. The subspaces of R2 are: 1. all of R2 , � 2. any line through 0 0 � and 3. the zero vector alone (Z). The subspaces of R3 are: 1. all of R3 , 2. any plane through the origin, 3. any line through the origin, and 4. the zero vector alone (Z). Column space Given a matrix A with columns in R3 , these columns and all their linear � combi � 3 3 , 1 � � 1 the column space of A is the plane through the origin in R3 containing 2 4 � � 3 and 3 . 1 nations form a subspace of R3 . This is the column space C ( A). If A = 1 2 4 Our next task will be to understand the equation Ax = b in terms of sub spaces and the column space of A.