Linear Study Set

Fertilizer is sold in bags labelled with the amount of nitrogen (N), phosphoric acid (P₂O₅), and potash (K₂O present. The mixture of these nutrients varies from one type of fertilizer to the next. For example, a bag of Vigoro Ultra Turf fertilizer contains 29 pounds of nitrogen, 3 pounds of phosphoric acid, and 4 pounds of potash. Another type of fertilizer, Parker's Premium Starter, has 18 pounds of nitrogen, 25 pounds of phosphoric acid, and 6 pounds of potash per bag. Determine the number of bags of each type required to yield a mixture containing 130 pounds of nitrogen, 106 pounds of phosphoric acid, and 32 pounds of potash.

The mixture contains 2 bags of Vigoro, and 4 bags of Parker's.

Determine the value of h such that the matrix is the augmented matrix of a consistent linear system. [6 −2|h] [−12 4|6]

h = -3

Determine all values of h and k for which the system {5 +9y = h -8x + ky = 1}

k = -72/5 h ≠ -5/8

True/False The matrix 1 −2 0 | 0 0 0 1 | 0 0 0 0 | 0 is in reduced row echelon form.

true

Lisa and Sam are sister and brother. Lisa has as many brothers as sisters, and Sam has twice as many sisters as brothers. How many girls and boys are there in this family?

4 girls 3 boys Let b be the number of boys in the family and let g be the number of girls. The sentence "Lisa has as many brothers as sisters" means b=g−1, and the sentence "Sam has twice as many sisters as brothers" means g=2(b−1). Hence this is a system of linear equations; its solution is: 4 girls and 3 boys.

True/false If an augmented matrix in reduced row echelon form has 2 rows and 3 columns (to the left of the vertical bar), then the corresponding linear system has infinitely many solutions.

False

True/false If the number of rows of an augmented matrix in reduced row echelon form is greater than the number of columns (to the left of the vertical bar), then the corresponding linear system has infinitely many solutions.

False

True/False A system of 4 linear equations in 5 variables can have exactly one solution.

False. The augmented matrix can have at most 4 pivots, so there will be at least one free variable, thus any associated system will either be inconsistent or have infinitely many solutfions

Ax = 0 is what type of system?

Homogeneous system

If a linear set has a pivot in every column then?

It is linear independent

Suppose that a system of two equations with six unknowns corresponds to a matrix in row echelon form. What is the largest possible number of pivots for this matrix?

Pivots: 2 Each equation corresponds to a row in the matrix, so the matrix has 2 rows (and 7 columns). There are at most 2 pivots since the each row contains at most one pivot. (It may have no pivot if the entire row is zero.)

True/false If the solution to a system of linear equations is given by (4−2z,−3+z,z) then (4,−3,0) is a solution to the system. Choose True False

True

True/Flase Suppose A is an 4 × 3 matrix whose first column is the sum of its second and third columns. Then the equation Ax = 0 has infinitely many solutions.

True. The columns of A are linearly dependent, thus Ax = 0 has infinitely many solutions.

True/False If A is an m × n matrix and m > n, then then there is at least one vector b in R m which is not in the span of the columns of A.

True. The matrix A has at most n pivots, but it has m rows and m > n so it cannot have a pivot in every row

True/False The vector equation x₁ 1 + x₂ 2 2 −1 -2 = 2 is consistent

c) False. The associated augmented matrix has a pivot in the rightmost column.

If a linear system has four equations and seven variables, then it must have infinitely many solutions.

false

Solve the system using row operations on an augmented matrix. −2x + 5y - 3z = 23 3x -2y + 4z = -27 −6x +3y −6z =45

x = -3 y = 1 z = -4

Solve the following linear system using row reduction: −2x -5y + 5z = 28 3x -4y + 2z = 17 -3x -5y -4z = -18

x =1, y= -1, z = 5