Name ____________________________________
Instructions: Please show all work in the spaces below, including showing and recording all steps for row reduction.
1. Fill-in the blank spaces or circle the appropriate answer.
a. Matrix is in row echelon form/reduced row echelon form/neither. [2 points]
i. Circle all pivot positions in the matrix. [2 points]
ii. List all basic variables __________ and free variables _________. [4 points]
iii. Is the equation consistent when ? Why? [4 points]
b. The set , where and is linearly _______________
because _________________________________________. [4 points]
2. Write TRUE or FALSE. Justify. [4 points each]
a. If equation has more than one solution, then equation also has more than one solution.
b. Suppose that a coefficient matrix for a system has four pivot columns, then the system is consistent.
c. The homogeneous equation has the trivial solution if and only if the equation has at least one free variable.
d. The columns of are linearly independent if the equation has the trivial solution.
e. If is a matrix, then the range of the transformation is .
3. Given the system .
a. Write the corresponding augmented matrix for the system, the row-reduce the matrix to row-echelon form. [4 points]
b. Find all the values of and such that the solution set of the system is
i. Empty. [4 points]
ii. Contains a unique solution. [4 points]
iii. Contains infinitely many solutions. [4 points]
4. Given matrix and vector , write equation
a. As a vector equation. [4 points]
b. As a system of linear equations. [4 points]
5. Consider the system associated to the augmented matrix . Write the general solution of the associated system in parametric vector form. [10 points]
6. Consider the matrix .
a. Row-reduce to reduced echelon form, recording all elementary row operations used.
[8 points]
b. Are the columns of linearly independent? Why or why not? [4 points]
c. Do the columns of span Justify. [4 points]
7. Let be a linear transformation such that , , , and . Find the image of by the transformation . [8 points]
8. The transformation is defined by Show that is not linear. [8 points]
Page 1 of 5
corvettec6
hhhhhhhhhhhhh.docx
0
