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Solutions to various problems related to systems of linear equations and matrices, including determining consistent linear systems, finding row reduced echelon forms, writing vector equations, and solving matrix equations. It also covers topics such as linear independence and parametric vector solutions.
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Determine when the augmented matrix represents a consistent linear system.
1 0 2 a 2 1 5 b 1 − 1 1 c
We need to reduce the given matrix:
1 0 2 a 2 1 5 b 1 − 1 1 c
− 1 R 1 + R 3 ÏR 3
1 0 2 a 0 1 1 b − 2 a 0 − 1 − 1 c − a
1 0 2 a 0 1 1 b − 2 a 0 0 0 b + c − 3 a
The given matrix represents a consistent linear system if and only if b + c − 3 a = 0.
Find the row reduced echelon form of the matrix below and mark the pivot positions: 1 − 2 − 4 3 2 5 − 2 9 1 7 2 6 0 5 − 2 9 1 − 2 − 4 3
We need to reduce the given matrix: 1 − 2 − 4 3 2 5 − 2 9 1 7 2 6 0 5 − 2 9 1 − 2 − 4 3
− 2 R 1 + R 2 ÏR 2 ,−R 1 + R 3 ÏR 3 −R 1 + R 4 ÏR 4
−R 2 + R 3 ÏR 3 R 3 ↔R 4
5 R 2 + R 3 ÏR 3 − 2 R 3 + R 1 ↔R 1
( − 1 / 2) R 3 + R 1 ÏR 1 , (1 / 6) R 3 ↔R 3 − 2 R 3 + R 1 ↔R 1
( − 10 / 8) R 3 + R 2 ÏR 2 , (1 / 8) R 3 ↔R 3 −R 2 ↔R 2
Consider the linear system
3 x 1 + 2 x 2 − x 3 − x 4 = − 3 −x 1 + x 3 + 2 x 4 = 1 2 x 1 + 2 x 2 + x 4 = − 2 x 1 + 2 x 2 + x 3 + 3 x 4 = − 1
A. Write the linear system in the matrix form A x = b.
x 1 x 2 x 3 x 4
B. Solve the matrix equation A x = b and write the solution in parametric-vector form.
We need to reduce the augmented matrix for the system above. We perform the following operations:
3 R 2 + R 1 ÏR 1 , 2 R 2 + R 3 ÏR 3 R 1 + R 4 ÏR 4
−R 1 + R 3 ÏR 2 ,−R 1 + R 4 ÏR 4 R 1 ↔R 2
−R 1 ÏR 1 (1 / 2) R 2 ↔R 2
+^ x^3
+^ x^4
:^ x^3 , x^4 ∈^ R
C. Give a particular solution for the matrix equation A x = b
The vector p =
is a particular solution.
D. Write the solution set for the matrix equation A x = 0.
v h =
x 3
+^ x^4
:^ x^3 , x^4 ∈^ R
Let v 1 =
, v 2 =
, v 3 =
(^) and v 4 =
A. Show that S = { v 1 , v 2 , v 3 , v 4 } is linearly dependent.
We need to reduce the matrix [v 1 v 2 v 3 v 4 ].
−R 1 + R 3 ÏR 3
2 R 3 + R 1 ↔R 1
(1 / 5) R 3 ↔R 3
− 2 R 3 + R 1 ↔R 1 ,−R 2 ↔R 2
Note that the vector equation xv 1 ⃗ (^) 1 + x 2 v⃗ (^) 2 + xv 3 ⃗ (^) 3 + x 4 v⃗ (^) 4 = 0 has infinitely many solutions because x 4 is a free variable. Therefore, {v⃗ (^) 1 v,⃗ (^) 2 v,⃗ (^) 3 ,⃗v (^) 3 } is linearly dependent.
B. Show that T = { v 1 , v 2 , v 3 } is linearly independent.
The calculation in part A. showed that xv 1 ⃗ 1 + x 2 v⃗ (^) 2 + xv 3 ⃗ (^) 3 = 0 has only the trivial solution. Therefore, { v 1 , v 2 , v 3 } is linearly independent.
C. Show that v 4 can be written as a linear combination of { v 1 , v 2 , v 3 }.
The calculation in part A. showed that (2 / 5) v⃗ (^) 1 + ( − 6 / 5) v⃗ (^) 2 + (9 / 5) v⃗ (^) 3 = v ⃗ 4. Therefore, v⃗ (^) 4 can be written as a linear combination of { v 1 , v 2 , v 3 }.
Mark each statement True or False. You don’t need to justify your answer. Let S be a set of m vectors in R n.
A.) If m > n then S is linearly independent. FALSE.
B.) If the zero vector is in S , then S is linearly dependent. TRUE.
C.) If S is linearly independent and T is a subset of S , then T is linearly independent. TRUE.
D.) If T is linearly dependent and T is a subset of S , then S is linearly dependent. TRUE.
E.) The linear system A x = b has a unique solution if and only if the column vectors of A are linearly independent. TRUE.
