Algebra Lecture Slides PDF

Algebra Associate Professor Nguyen Trung Thanh Faculty of Data Science and Artificial Intelligence Rank of matrix 2/13 Column space Definition (Column space) Let A be an m × n matrix. The m−tuples corresponding to the columns of A are called column vectors of A. The column spa...

Algebra Associate Professor Nguyen Trung Thanh Faculty of Data Science and Artificial Intelligence Rank of matrix 2/13 Column space Definition (Column space) Let A be an m × n matrix. The m−tuples corresponding to the columns of A are called column vectors of A. The column space, denoted by col(A), of A is the subspace of Rm spanned by the column vectors of A. Examples       1 3 1 3 A =  2 2 has two column vectors x1 = 2 , and x2 = 2. 3 0 3 0 col(A) = Span{x1 , x2 } is a subspace of R3. 3/13 Row space Definition (Row space) Let A be an m × n matrix. The n−tuples corresponding to the rows of A are called row vectors of A. The row space, denoted by row(A), of A is the subspace of Rn spanned by the rows of A. Examples   1 3 [ ] [ ] [ ] 1 2 3 A =  2 2 has three row vectors y1 = , and y2 = , and y3 =. 3 2 0 3 0 row(A) = Span{y1 , y2 , y3 } is a subspace of R2. 4/13 Properties Property 1 Let A and B be m × n matrices. If A → B by elementary row operations, then row(A) = row(B). If A → B by elementary column operations, then col(A) = col(B). Property 2 If R is a row-echelon matrix, then The nonzero rows of R are a basis of row(R). The columns of R containing leading ones are a basis of col(R). 5/13 Properties Property 3 Suppose A is an m × n matrix. Then the row space and column space of A have same dimension. Examples   1 3 A =  2 2. 3 0 Verify that dim(row(A)) = dim(col(A)). 6/13 Rank of matrix Definition Let A be an m × n matrix. The dimension of the row space (equivalently, of the column space) of A is called the rank of A and is denoted by rank(A). dim(row(A)) = dim(col(A)) = rank(A) Examples Find the rank of the matrix   1 3 A =  2 2. 3 0 7/13 Properties Property Let A, B be m × n matrices, and C be n × p matrix. 1. rank(A) = rank(A⊤ ) 2. The rank of a null matrix is zero 3. The rank of an identity matrix of order n is n 4. rank(A) ≤ min{m, n} 5. rank(A) = rank(AU) = rank(VA) whenever U and V are invertible 6. If A → B by elementary row/column operations, then rank(A) = rank(B). 7. rank(A + B) ≤ rank(A) + rank(B) 8. rank(AC) ≤ min{rank(A), rank(C)} 8/13 Null space of matrix Definition Suppose A is an m × n matrix. The null space of A, denoted by null(A), is the set of solutions of the homogeneous system Ax = 0. Notationally: null(A) = {x ∈ Rn | Ax = 0} The dimension of null(A) is called the nullity of A. Examples Find the null space of the matrix [ ] 1 3 1 A=. 0 2 −1 9/13 Image space of matrix Definition Suppose A is an m × n matrix. The image space of A, denoted by im(A), is defined as: im(A) = {Ax | x ∈ Rn } Examples Find the image space of the matrix [ ] 1 3 1 A=. 0 2 −1 10/13 Properties Property Let A be an m × n matrix. 1. The n − r basic solutions to the system Ax = 0 provided by the Gaussian algorithm are a basis of null(A), so dim(null(A)) = n − r 2. dim(im(A)) = r Examples Find bases of the null and image spaces (and their dimension) of the matrix   1 −2 1 1 A = −1 2 0 1. 2 −4 1 0 11/13 Other properties Property The following are equivalent for an m × n matrix A. 1. rank(A) = n 2. The rows of A span Rn 3. The columns of A are linearly independent in Rm 4. The matrix A⊤ A is invertible 5. CA = I for some n × m matrix C 6. If Ax = 0, x ∈ Rn then x = 0 7. If m = n, then det(A) ̸= 0 12/13 Other properties Property The following are equivalent for an m × n matrix A. 1. rank(A) = m 2. The columns of A span Rm 3. The rows of A are linearly independent in Rn 4. The matrix AA⊤ is invertible 5. AC = I for some n × m matrix C 6. The system Ax = b is consistent for every b ∈ Rm 7. If m = n, then det(A) ̸= 0 13/13