Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications The Relationship between Rank and Nullity A Fundamental Theorem for Linear Algebra A. Havens Department of Mathematics University of Massachusetts, Amherst March 28, 2018 A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Outline 1 Rank, Nullity, and The Row Space Review: Column Space and Null Space Rank and Nullity Row Space 2 The Rank-Nullity Theorem The Theorem for Matrices A Fundamental Theorem of Linear Algebra 3 Interpretation and Applications The Rigidity of Linear Maps Left Null Spaces Rank One Matrices and Outer Products Duality and the Row Space A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Review: Column Space and Null Space Definitions of Column Space and Null Space Definition m×n Let A 2 R be a real matrix. Recall m The column space of A is the subspace Col A of R spanned m by the columns of A: Col A = Span fa1;:::; ang ⊆ R where î ó A = a1 ::: an . n m Equivalently, Col A is the same as the image T (R ) ⊆ R of the linear map T (x) = Ax. n The null space of A is the subspace Nul A of R consisting of all vectors x which are solutions to the homogeneous equation n n with matrix A: Nul A := fx 2 R j Ax = 0g ⊆ R Equivalently, it is the kernel of the map T (x) = Ax. A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Rank and Nullity Rank: The Dimension of the Column Space Definition The rank of a linear map T : V ! W between finite dimensional vector spaces V and W is the dimension of the image: rank T = dim T (V ) : Given an m × n matrix A, the rank of A is the dimension of the column space of A: rank A = dim Col A : Remark Observe that rank T ≤ dim V and rank T ≤ dim W . A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Rank and Nullity Remark We know rank T ≤ dim V because the image subspace is spanned by the images of basis vectors, and so in particular, T (V ) is spanned by a set of dim V vectors, which is an upper bound on the size of a linearly independent spanning set. That rank T ≤ dim W follows from the fact that T (V ) is a subspace of W , and so its dimension is less than or equal to the dimension of W . A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Rank and Nullity Nullity: The Dimension of the Null Space Definition The nullity of a linear map T : V ! W between finite dimensional vector spaces V and W is the dimension of the kernel: nullity T = dim ker T : Given an m × n matrix A, the nullity of A is the dimension of the null space of A: nullity A = dim Nul A : A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Rank and Nullity Remark Observe that nullity T ≤ dim V , but it need not be bounded by the dimension of W . Exercise Explain the above remark about the bound on the nullity of a linear map. What is the relationship between the dimension of the codomain and the nullity? A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Rank and Nullity Finding a Basis of the Column Space To find a basis of the Column space of A, find a row equivalent matrix B is echelon form and locate the pivot columns. Recall that the corresponding columns of A are the pivot columns of A. As each non-pivot column corresponds to a free variable, any non-pivot columns may be realized as linear combinations of the pivot columns. Thus the pivot columns of A are a maximal linearly independent subset of the column of A, and span Col A. That is, the pivot columns of A are a basis of the column space of A. A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Rank and Nullity Finding a Basis of the Null Space To find a basis of the null space of A, solve the homogeneous system Ax = 0. The solution vector x can be written as a linear combination of some vectors weighted by free variables. Since each such vector corresponds to a unique free variable, it will have a one in a coordinate position where the other vectors have zeros. Thus, the matrix whose columns are these vectors spanning the null space has as many pivots as the system has free variables, and so this collection is linearly independent and forms a basis of the null space. A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Rank and Nullity Summarizing: Computing Rank and Nullity A basis of Col A is given by the pivot columns of A. Be careful to use the columns of the original matrix A and not of RREF(A)! The rank is the number of pivot columns of A, or equivalently the number of pivot positions of RREF(A). A basis of Nul A is found by solving the homogenous equation and decomposing a parameterized solution vector as a linear combination of vectors weighted by free variables. The vectors in this sum form a basis of Nul A. The nullity of A is thus the number of free variables for the homogeneous system, which is the same as the number of non-pivot columns of A. A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Row Space Definition of the Row Space of a Matrix Definition Let A be an m × n matrix. The set of all linear combinations of the rows of A is called the row space of A. Since each row of A is a column of At, we often regard the row t m space as Col A ⊆ R , though strictly speaking, these are m isomorphic but not equal spaces, and Row A 6⊆ R , as it consists of objects distinct from column vectors. Later, we'll define the correct space within which the row space is realized as a subspace. A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Row Space Row Equivalence and Row Spaces Theorem Suppose A ∼ B. Then Row A = Row B. Moreover, if B is in echelon form, then the nonzero rows of B form a nonzero basis of both Row B and Row A. Proof. Suppose B is obtained from A by row operations. Then the rows of B are linear combinations of the rows of A, and thus any linear combination of rows of B is a linear combination of rows of A, whence Row B ⊆ Row A. Since elementary row operations are invertible, we can argue symmetrically that Row A ⊆ Row B, whence Row A = Row B. A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Row Space Proof (continued.) If B is in echelon form, then the nonzero rows form a linearly independent set, as no nonzero row can be made as a linear combination of the nonzero rows below it. Equivalently, by transposing, consider that each nonzero row has a pivot position, and gives a pivot column of Bt. Thus the nonzero rows of B form a basis of Row B = Row A. A. Havens The Relationship between Rank and Nullity Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Row Space An Example Example Find bases of the row space, column space, and null space of the matrix 2 1 4 3 2 5 3 6 4 8 12 9 0 7 6 7 A = 6 7 : 4 3 4 9 7 −5 5 2 8 6 5 6 A. Havens The Relationship between Rank and Nullity R3 − 3R1 7! R3, R4 − 2R1 7! R3, R3 − R2 7! R3, and R3 $ R4, we obtain a row equivalence 2 1 4 3 2 5 3 6 0 −8 0 1 −20 7 6 7 A ∼ B = 6 7 4 0 0 0 1 −4 5 0 0 0 0 0 Since B is in echelon form, we deduce that î ó 8 1 4 3 2 5 ; 9 <> î ó => Row A = Span 0 −8 0 1 −20 ; : î ó :> 0 0 0 1 −4 ;> Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Row Space Example Using the row operations R2 − 4R1 7! R2, A. Havens The Relationship between Rank and Nullity R4 − 2R1 7! R3, R3 − R2 7! R3, and R3 $ R4, we obtain a row equivalence 2 1 4 3 2 5 3 6 0 −8 0 1 −20 7 6 7 A ∼ B = 6 7 4 0 0 0 1 −4 5 0 0 0 0 0 Since B is in echelon form, we deduce that î ó 8 1 4 3 2 5 ; 9 <> î ó => Row A = Span 0 −8 0 1 −20 ; : î ó :> 0 0 0 1 −4 ;> Rank, Nullity, and The Row Space The Rank-Nullity Theorem Interpretation and Applications Row Space Example Using the row operations R2 − 4R1 7! R2, R3 − 3R1 7! R3, A.