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ECS423U (2019) Page 3 Question 2 (Determinants and Vector Spaces) a) Consider the following system of...
2 x [b] Consider the following linear system of equations AX =B : (i) Determine a basis for the row space of A. (ii) Compute the Rank of the augmented matrix (A:B), then use it to classify the solution of this system (Unique - Many -No: solution). (iii) Is the matrix A diagonalizable? Explain your answer and verify the similarity transformation.
[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]
1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all symmetric 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, 2a +36) (which is a subspace of Re"). 2. (12 pts) Given the matrix in a R R-E form: -21 1 [1 0 0 0 3 0 1 1 0 - 2 0 0 0 1 0...
Determine which property of determinants the equation illustrates. 1 3 2 0 0 0 96 -8 = 0 If one row of a matrix is a multiple of another row, then the determinant of the matrix is zero. If one row of a matrix consists entirely of zeros, then the determinant of the matrix is zero. If two columns of a matrix are interchanged, then the determinant of the matrix changes sign. If a row of a matrix is multiplied...
3. Consider the following system of linear equations: 2x + 2y + 2kz = 2 kx + ky+z=1 2x + 3y + 7z = 4 (i) Turn the system into row echelon form. (ii) Determine which values of k give (i) a unique solution (ii) infinitely many solutions and (iii) no solutions. Show your working. 2. Let v= [6, 1, 2], w = [5,0, 3), and P= (9, -7,31). (i) Find a vector u orthogonal to both v and w....
7. This question involves the concept of determinants and partitioned matrices. Historically, determinants first arose in the context of solving systems of linear equations for one set of variables in terms of another. For example, if the coefficient matrix of the system u= ax + by v=cx + dy is invertible, then the equations can be solved for x and y in terms of u and v as au – cu 2= du - bv ad - bc y =...
1. Consider the matrix 12 3 4 A 2 3 4 5 3 4 5 6 As a linear transformation, A maps R' to R3. Find a basis for Null(A), the null space of A, and find a basis for Col(A), the column space of A. Describe these spaces geometrically. 2. For A in problem 1, what is Rank(A)?
5. For the system, 4x + y + 2z = 1 2x + 3y + 4z = -5 x – y +3z = 3 Find the rank of the coefficient matrix by calculating the determinant. Use Cramer's theorem to find the solution of this system. (10 points) 6. Find the inverse of the following matrix using Gauss-Jordan method. Verify your result by computing the inverse using the method of determinants. (10 points) 1 2 4 2 4 2 1] 1...
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A using cofactor expansion of row 3. (ii) Is A invertible? If so, give the third column of A1 (you do not have to simplify any fractions) (b) Let B be the matrix 0 0 4 0 2 8 0 4 2 1 0 0 0 7 Use row operations to find the determinant of B. Make...
U 10 I LI IU I LILIUI IUL 2. (Bases for column and row spaces) Suppose A is a 3 x 3 matrix with pivots in exactly the first and second columns. Suppose B is the unique reduced echelon form of A. (a) We know that the first and second columns of A form a basis for Col A. Is it necessarily true that the first and second columns of B form a basis for Col A? If so, explain...