Linear Algebra: The Matrix Equation:

Q1.

Given A and b to the rig ht, write the augmented matrix for the linear system that corresponds to the matrix equation Ax = h. Then solve the system and write the A = 2 5 1 , b = 20 solution as a vector. 3 2 6 16 E) Write the augmented matrix for the linear system that corresponds to the matrix equation Ax = b. Select the correct choice below and fill in any answer boxes within your choice. Let u = 1 and A = - 5 7 . ls u in the plane in R3 spanned by the columns of A? Why or why not? 10 2 2 (3 Select the correct choice below and ll in the answer box to complete your choice. (Type an integer or decimal for each matrix element.) '31::3' A- No, the reduced row echelon form of the augmented matrix is , which is an inconsistent system. 'C?' 3- Yes, multiplying A by the vector writes u as a linear combination of the columns of A. How many rows of A contain a pivot position? Does the equation Ax = b have a solution for each b in [F 2 4 'i 2 -2 -2 -2 4 0 -6 3 -18 2 0 3 -8 A: How many rows of A contain a pivot position? A has D rows which contain a pivot position. Can each vector in R4 be written as a linear combination of the columns of the matrix A? Do the columns ofAspan R4? 3 1 6 -9 -1 -1 -1 1 O -2 3 -6 2 0 5 -13 A: E) Can each vector in [R4 be written as a linear combination of the columns of the matrix A? Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal for each matrix element.) {3. A. Yes, because the reduced row echelon form of A is {:1} B. No, because the reduced row echelon form of A is O - 1 0 Let v1 = 0 , v2 = 0 , and v3 = 1 . Does {v1 ,v2,v3} span R4? Why or why not? - 1 0 - 1 Choose the correct answer below. {:3- A. No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only three rows. {:2- B. Yes. When the given vectors are written as the columns of a matrix A, A has a pivot position in every row. '3' C- Yes. Any vector in [R4 except the zero vector can be written as a linear combination of these three vectors. {3' D- No. The set of given vectors spans a plane in R4. Any of the three vectors can be written as a linear combination of the other two. Determine whether each statement below is true or false. Justify each answer. @- a. The equation Ax = b is referred to as a vector equation. Choose the correct answer below. {:1- A. False. The equation Ax = b is referred to as a linear equation because b is a linear combination of vectors. {:2} B. True. The equation A): = b is referred to as a vector equation because it consists of scalars multiplied by vectors. {:1- C. False. The equation Ax = b is referred to as a matrix equation because A is a matrix. {:2- D. True. The equation Ax = b is referred to as a vector equation because A is constructed from column vectors. - 5 5 - 50 Let u = 2 V= - 5 , and w = 41 . It can be shown that 3u - 7v - w = 0. Use this fact (and no row operations) to find x, and - 2 29 - 5 5 - 50 X1 X2 that satisfy the equation 2 - 5 = 41 5 - 2 29 . . . X2 (Simplify your answers.)