Properties of Linear Transformations, Matrices 1

Math 4377/6308 Advanced Linear Algebra 2.2 Properties of Linear Transformations, Matrices. Math 4377/6308, Advanced Linear Algebra Spring, 2015 1/1 2.2 Properties of Linear Transformations, Matrices. Null Spaces and Ranges Injective, Surjective, and Bijective Dimension Theorem Nullity and Rank Linear Map and Values on Basis Coordinate Vectors Matrix Representations Math 4377/6308, Advanced Linear Algebra Spring, 2015 2/1 Linear Map and Null Space Theorem (2.1-a) Let T : V → W be a linear map. Then null(T ) is a subspace of V . Math 4377/6308, Advanced Linear Algebra Spring, 2015 3/1 Injective, Surjective, and Bijective Linear Maps Definition The linear map T : V → W is called injective (one-to-one) if, for all u, v ∈ V , the condition Tu = Tv implies that u = v . In other words, different vectors in V are mapped to different vectors in W . Definition The linear map T : V → W is called surjective (onto) if range(T ) = W . Definition A linear map T : V → W is called bijective if T is both injective and surjective. Math 4377/6308, Advanced Linear Algebra Spring, 2015 4/1 Injective Linear Map Theorem (2.4) Let T : V → W be a linear map. Then T is injective if and only if null(T ) = {0}. Math 4377/6308, Advanced Linear Algebra Spring, 2015 5/1 Linear Maps and Ranges Theorem (2.1-b) Let T : V → W be a linear map. Then range(T ) is a subspace of V. Math 4377/6308, Advanced Linear Algebra Spring, 2015 6/1 Dimension Theorem Theorem (2.3, Dimension Theorem) Let V be a finite-dimensional vector space and T : V → W be a linear map. Then range(T ) is a finite-dimensional subspace of W and dim(V ) = dim(null(T )) + dim(range(T )). Math 4377/6308, Advanced Linear Algebra Spring, 2015 7/1 Dimension Theorem (cont.) Math 4377/6308, Advanced Linear Algebra Spring, 2015 8/1 Dimension Theorem (cont.) Math 4377/6308, Advanced Linear Algebra Spring, 2015 9/1 Surjective Linear Map Corollary Let T : V → W be a linear map. 1 If dim(V ) > dim(W ), then T is not injective. 2 If dim(V ) < dim(W ), then T is not surjective. Math 4377/6308, Advanced Linear Algebra Spring, 2015 10 / 1 Nullity and Rank Definition For vector spaces V , W and linear T : V → W , if null(T ), i.e., N(T ), and range(T ), i.e, R(T ), are finite-dimensional, the nullity and the rank of T are the dimensions of null(T ) and range(T ), respectively. Theorem (Dimension Theorem, 2.3) For vector spaces V , W and linear T : V → W , if V is finite-dimensional then nullity(T ) + rank(T ) = dim(V ) Math 4377/6308, Advanced Linear Algebra Spring, 2015 11 / 1 Rank of a Matrix Rank The rank of A is the dimension of the column space of A. rank A = dim Col A =# of pivot columns of A = dim Row A . The set of all linear combinations of the row vectors of a matrix A is called the row space of A and is denoted by Row A. Col AT = Row A . Note the following: dim Col A = # of pivots of A = dim Row A. dim Nul A = # of free variables = # of nonpivot columns of A. Math 4377/6308, Advanced Linear Algebra Spring, 2015 12 / 1 Rank Theorem |rank {z A} l   # of pivot   columns   of A + Nul A} |dim {z l   # of nonpivot   columns   of A = n |{z} l   # of   columns   of A Theorem (Rank Theorem) The dimensions of the column space and the row space of an m × n matrix A are equal. This common dimension, the rank of A, also equals the number of pivot positions in A and satisfies the equation rank A + dim Nul A = n. Math 4377/6308, Advanced Linear Algebra Spring, 2015 13 / 1 Rank Theorem: Example Since Row A = Col AT , rank A = rank AT . Example Suppose that a 5 × 8 matrix A has rank 5. Find dim Nul A, dim Row A and rank AT . Is Col A = R5 ? Solution: Nul A} = |{z} n rank | {z | {z A} + dim l ↓ l 5 ? 8 5 + dim Nul A = 8 ⇒ dim Nul A = dim Row A = rank A = ⇒ rank AT = rank = Since rank A = # of pivots in A = 5, there is a pivot in every row. So the columns of A span R5 . Hence Col A = R5 . Math 4377/6308, Advanced Linear Algebra Spring, 2015 14 / 1 Rank Theorem: Example Example For a 9 × 12 matrix A, find the smallest possible value of dim Nul A. Solution: rank A + dim Nul A = 12 dim Nul A = 12 − rank | {z A} largest possible value= smallest possible value of dim Nul A = Math 4377/6308, Advanced Linear Algebra Spring, 2015 15 / 1 Properties of Linear Transformations Theorem (2.5) For vector spaces V , W of equal (finite) dimension and linear T : V → W , the following are equivalent: (a) T is one-to-one. (b) T is onto. (c) rank(T ) = dim(V ) Math 4377/6308, Advanced Linear Algebra Spring, 2015 16 / 1