Math 324: Linear Algebra Section 4.6: Row and Column Spaces Mckenzie West Last Updated: November 7, 2019 2 Last Time. { Pruning or extending for a basis { Basis for and Dimension of a Subspace Today. { Row and Column Spaces { Rank of a Matrix 3 Exercise 1. Consider the subspace of P3, given by 2 3 W = a + bx + ax + bx : a; b 2 R : (a) Show that every \vector" p(x) = a + bx + ax2 + bx3 can be written as a linear combination of u(x) = 1 + x2 and v(x) = x + x3: (b) Show that S = fu(x); v(x)g is linearly independent. (c) Deduce that S is a basis for W and that dim(W ) = 2. 4 Definition. Let A be an m × n matrix. 1. The column space of A, denoted col(A), is the subspace of m R that is spanned by the columns of A. n 2. the row space of A, denoted row(A), is the subspace of R that is spanned by the rows of A. Example. 28 −1 7 03 For A = 40 −1 −1 35, 0 −3 −3 9 3 { col(A) is a subspace of R because each column has 3 entries 4 { row(A) is a subspace of R because each row has 4 entries 5 Theorem 4.13. If an m × n matrix A is row-equivalent to an m × n matrix B, then row(A) = row(B). Theorem 4.14. If a matrix A is row-equivalent to a B that is in row-echelon form, then the non-zero rows of B form a basis for row(A). Exercise 2. 28 −1 7 03 Let A = 40 −1 −1 35. 0 −3 −3 9 Find a basis for row(A). What is the dimension of row(A)? 6 Exercise 3. 28 −1 7 03 Let A = 40 −1 −1 35. 0 −3 −3 9 Consider the matrix AT . (a) Convince yourself that col(A) = row(AT ). (b) Find a basis for row(AT ) using Theorem 4.14. (c) What is a basis for and the dimension of col(A)? 7 Exercise 4. Find a basis for col(A) and row(A) for 2 5 1 −9 −5 2 3 6 2 8 4 −2 9 7 A = 6 7 4 3 −5 −11 −3 −35 −3 2 8 3 7 8 Exercise 5. Find a basis for col(A) and row(A) for 2 7 −53 6 2 7 7 A = 6 7 4−2 3 5 9 9 9 Theorem 4.15. If A is an m × n matrix, then the row space and the column space of A have the same dimension. Exercise 6. Convince yourself of this fact by considering the relationship between dimension of row/column spaces and leading 1's in the reduced row echelon form of A. 10 Definition. The rank of a matrix A, denoted rank(A), is the dimension of the row (or column) space of A. Example. rank(In) = n rank(Om;n) = 0 11 Exercise 7. Determine 027 9 −131 B61 −2 −27C rank B6 7C @44 0 −15A 7 −3 −2 12 Exercise 8. Give examples of matrices such that (a) rank(A + B) < rank(A) and rank(A + B) < rank(B) (b) rank(A + B) = rank(A) and rank(A + B) = rank(B) (c) rank(A + B) > rank(A) and rank(A + B) > rank(B) Is it possible for rank(A + B) < rank(A) and rank(A + B) > rank(B)? 13 Exercise 9. Consider the homogeneous system with coefficient matrix 2 5 1 −9 −5 2 3 6 2 8 4 −2 9 7 A = 6 7 : 4 3 −5 −11 −3 −35 −3 2 8 3 7 (a) Solve the system. n (b) Verify that the set of solutions is a subspace of R for some n. (Also determine the value of n.) (c) What is a basis for and the dimension of the solution set? 14 Exercise 10. Consider the homogeneous system with coefficient matrix 2 7 −53 6 2 7 7 A = 6 7 4−2 3 5 9 9 (a) Solve the system. n (b) Verify that the set of solutions is a subspace of R for some n. (Also determine the value of n.) (c) What is a basis for and the dimension of the solution set? 15 Definition. If A is an m × n matrix, the null space of A, denoted null(A), is n the subspace of R consisting of the solutions to A~x = ~0. Note. We will begin next time by proving that null(A) is indeed a n subspace of R ..