[15 marks] b) Consider the following matrix: IT 2 0 -17 A = |2 6 -3 -38 3 10 -6 -5| i) Find the rank of A ii) Find a basis for the null-space of A iii) Find a basis for the row space of A [10 marks]
2. Let [8 Marks] 1 2 -1 1 3 -2 a) Find the null space of the matrix A and determine its dimension b) Find the range of the matrix A and determine rank(A) c) State the rank-nullity theorem and verify that it is valid for the matrix A.
2. Let [8 Marks] 1 2 -1 1 3 -2 a) Find the null space of the matrix A and determine its dimension b) Find the range of the matrix A...
4. (8 marks) Let V be the vector space of solutions to the ODE y" hyperbolic functions y 0, spanned by the cosh r and y2 = sinh r, and let z1 = e and z2 = e = (a) Show that 21, %2} is a basis for V {1, 2to {yı, Y2}. Show all working (b) Find the transition matrix from the basis 3
4. (8 marks) Let V be the vector space of solutions to the ODE y"...
IV. Let (10 -3 2 A= 0 1 -54 3 -2 1 -2 (a) Find a basis for the null-space of A. (b) Find a basis for the column-space of A. We were unable to transcribe this image
(21) (15 marks). Given 1 A 1 3 1 3 4 0 0 1 1 0 0 2 2 2 0 0 3 3 3 (a) (5 marks). Find a basis for N(A) (null space of A). (b) (5 marks). Find the rank of A; (c) (5 marks). Find a basis for the column space of A.
eclass.srv.ualberta.ca 2 of 2 1. Consider the matrix 3-2 1 4-1 2 3 5 7 8 (a) Find a basis B for the null space of A. Hint: you need to verify that the vectors you propose 20 actually form a basis for the null space. (Recall: (1) the null space of A consists of all x e R with Ax = 0, and (2) the matrix equation Ax = 0 is equivalent to a certain system of linear equations.)...
1 3 -2 -5 2 11 1. Let A= 3 9 -5 -13 6 3 1 -2 -6 8 18 -1 -1 (a) Find a basis for the row space of A, i.e. Row(A). (b) Find a basis for the column space of A, i.e. Col(A). (c) Find a basis for the null space of A, i.e. Null(A). (d) Determine rankA and dim(Null(A)).
[1 2 0 1] 10. Let A 2 3 1 1 13 5 1 2 (a). Find the reduced row echelon form of A. (b). Using the answer for (a), find rank(A), and find a basis for Col(A). 11. Let A= Find a matrix P such that P-1AP is a diagonal matrix,
1 2 -3 1 -6 -2 5 2. 4. (10 points) Let A = (a) (5 points) Find a basis for col(A) and calculate rank(A). (b) (5 points) Find a basis for null(A) and calculate nullity(A).
2. Let A be the matrix [i 3 4 51 0 A= 1 1 1 | 1 2 -4 -5 -3 -3 -2 -1 (a) Find a basis of the column space. Find the coordinates of the dependent columns relative to this basis. (b) What is the rank of A? (c) Use the calculations in part (a) to find a basis for the row space.