1. Consider the system of equations: 2x1 x2 + 23 = 2 23x3 = 6 2x1 x2 + 7x3 = 14 (a) Write down the augmented matrix corresponding to this system. What is the size of the augmented matrix? (b) Find the echelon form of the augmented matrix. Underline the pivot positions. How many solutions will the system have? (c) Find the reduced echelon form. (d) Write the general solution in terms of the free variables. (The parametric vector form). 2. Consider the vector equation: 5 x1 2 0 +22 +3 (a) Write the system of equations that is equivalent to the vector equation. (b) Determine if there is a solution to the system, i.e., determine if the vector on the right-hand side is a linear combination of those of the left. (c) For what value of k is 2 in the span of k 5 -6 0 8 3. Compute the following products using the indicated method. 1 2 3 (a) -3 3 " (linear combination of columns) 4 5 (b) -3 " (row-column rule) 4 14 -22 (c) -3 3 (your preferred way) 14 9 4. Consider the system of equations. 4x1 -2 I2 + 3x3 = 3 -2x2 + 3x3 = -4 31 12 + 4x3 = 6 (a) Rewrite it as a vector equation. (b) Rewrite it as a matrix equation. 5. Let T be defined by T(x) = Ax. Find a vector x whose image under T is b and determine whether x is unique. 1 -2 3 A = 0 1 2 -5 -3,b= 6 -5 6. Show that the transformation T defined by T(x1,x2) = (x1 - 2x2,1 - 3,2x1 - 5x2) is not linear. 7. Consider the linear transformation T: R³ R² given by T(x1, x2, 13) = (x1 - 3x2 + 2x3,-x2+5x3). (a) Find a matrix that implements the mapping. (b) Is the linear transformation one-to-one? (c) Is the linear transformation onto? Chapter 2 1. Let A = [1 (a) Find A using the row reduction algorithm in example 7 (section 2.2 of the textbook). (b) Find A¹ using Theorem 4 in section 2.2 of the textbook. Make sure your answer is the same as above. (c) Solve the system x1 +4x2 = 2x14x2 2 == 3 using the matrix inverse you found.