# linear algebra homework

**Math 2240 Exam1 Fall 2015**

**Name** ____________________________________

**Instructions**: Please show all work in the spaces below, including showing and recording all steps for row reduction.

- Fill-in the blank spaces or circle the appropriate answer.
- Matrix is in row echelon form/reduced row echelon form/neither. [2 points]
- Circle all pivot positions in the matrix. [2 points]

- Matrix is in row echelon form/reduced row echelon form/neither. [2 points]

- List all basic variables __________ and free variables _________. [4 points]

- Is the equation consistentwhen ? Why? [4 points]

- The set , where and is linearly _______________

because _________________________________________. [4 points]

- Write TRUE or FALSE. Justify. [4 points each]
- If equation has more than one solution, then equation also has more than one solution.

- Suppose that a coefficient matrix for a system has four pivot columns, then the system is consistent.

- The homogeneous equation has the trivial solution if and only if the equation has at least one free variable.

- The columns of are linearly independent if the equation has the trivial solution.

- If is a matrix, then the range of the transformation is .

- Given the system .
- Write the corresponding augmented matrix for the system, the row-reduce the matrix to row-echelon form. [4 points]

- Findall the values of and such that the solution set of the system is
- Empty. [4 points]

- Contains a unique solution. [4 points]

- Contains infinitely many solutions. [4 points]

- Given matrix and vector , write equation
- As a vector equation. [4 points]

- As a system of linear equations. [4 points]

- Consider the system associated to the
**augmented matrix**.Write the general solution of the associated system in parametric vector form. [10 points]

- Consider the matrix .
- Row-reduce to reduced echelon form, recording all elementary row operations used.

[8 points]

- Are the columns of linearly independent? Why or why not? [4 points]

- Do the columns of span [4 points]

- Let be a linear transformation such that , , , and . Find the image of by the transformation . [8 points]

- The transformation is defined by Show that is not linear. [8 points]

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