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 ∈ 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 = {u(x), v(x)} 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. 8 −1 7 0 For A = 0 −1 −1 3, 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. 8 −1 7 0 Let A = 0 −1 −1 3. 0 −3 −3 9 Find a basis for row(A). What is the dimension of row(A)? 6
Exercise 3. 8 −1 7 0 Let A = 0 −1 −1 3. 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
5 1 −9 −5 2 2 8 4 −2 9 A = 3 −5 −11 −3 −3 −3 2 8 3 7 8
Exercise 5. Find a basis for col(A) and row(A) for
7 −5 2 7 A = −2 3 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 7 9 −1 1 −2 −2 rank 4 0 −1 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
5 1 −9 −5 2 2 8 4 −2 9 A = . 3 −5 −11 −3 −3 −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
7 −5 2 7 A = −2 3 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 .