Question

Give augmented matrix for this system. Find all solutions to this system. Indicate all parameters.

x1-x2+x3+x4=1

2x2+3x3+4x4=2

x1-x2+2x3+3x4=3

x1=? x2=? x3=? x4=?

Answer #1

system is

augmented matrix is

1 | -1 | 1 | 1 | 1 |

0 | 2 | 3 | 4 | 2 |

1 | -1 | 2 | 3 | 3 |

convert into Reduced Row Eschelon Form...

Add (-1 * row1) to row3

1 | -1 | 1 | 1 | 1 |

0 | 2 | 3 | 4 | 2 |

0 | 0 | 1 | 2 | 2 |

Divide row2 by 2

1 | -1 | 1 | 1 | 1 |

0 | 1 | 3/2 | 2 | 1 |

0 | 0 | 1 | 2 | 2 |

Add (-3/2 * row3) to row2

1 | -1 | 1 | 1 | 1 |

0 | 1 | 0 | -1 | -2 |

0 | 0 | 1 | 2 | 2 |

Add (-1 * row3) to row1

1 | -1 | 0 | -1 | -1 |

0 | 1 | 0 | -1 | -2 |

0 | 0 | 1 | 2 | 2 |

Add (1 * row2) to row1

1 | 0 | 0 | -2 | -3 |

0 | 1 | 0 | -1 | -2 |

0 | 0 | 1 | 2 | 2 |

reduced system is

general solution is

Linear Algebra
find all the solutions of the linear system using Gaussian
Elimination
x1-x2+3x3+2x4=1
-x1+x2-2x3+x4=-2
2x1-2x2+7x3+7x4=1

Find the number of solutions to
x1+x2+x3+x4=16 with
integers x1 ,x2, x3, x4
satisfying
(a) xj ≥ 0, j = 1, 2, 3, 4;
(b) x1 ≥ 2, x2 ≥ 3, x3 ≥ −3,
and x4 ≥ 1;
(c) 0 ≤ xj ≤ 6, j = 1, 2, 3, 4

x1-5x2+x3+3x4=1
2x1-x2-3x3-x4=3
-3x1-3x3+7x3+5x4=k
1 ) There is exactly one real number k for which the system has
at least one solution; determine this k and describe all solutions
to the resulting system.
2 ) Do the solutions you found in the previous part form a
linear subspace of R4?
3 ) Recall that a least squares solution to the system of equations
Ax = b is a vector x minimizing the length |Ax=b| suppose that
{x1,x2,x3,x4} = {2,1,1,1}
is a...

Find the standard matrix for the following transformation T : R
4 → R 3 : T(x1, x2, x3, x4) = (x1 − x2 + x3 − 3x4, x1 − x2 + 2x3 +
4x4, 2x1 − 2x2 + x3 + 5x4) (a) Compute T(~e1), T(~e2), T(~e3), and
T(~e4). (b) Find an equation in vector form for the set of vectors
~x ∈ R 4 such that T(~x) = (−1, −4, 1). (c) What is the range of
T?

Solve the following system of linear equations: 3x2−9x3 = −3
x1−2x2+x3 = 2 x2−3x3 = 0 If the system has no solution, demonstrate
this by giving a row-echelon form of the augmented matrix for the
system. If the system has infinitely many solutions, your answer
may use expressions involving the parameters r, s, and t. You can
resize a matrix (when appropriate) by clicking and dragging the
bottom-right corner of the matrix.

2. Find the number of integer solutions to x1 + x2 + x3 + x4 +
x5 = 50, x1 ≥ −3, x2 ≥ 0, x3 ≥ 4, x4 ≥ 2, x5 ≥ 12.

solve the following linear system by gauss-jordan
method
x1 + x2 - 2x3 + x4 = 8
3x1 - 2x2 - x4 = 3
-x1 + x2 - x3 + x4 = 2
2x1 - x2 + x3 - 2x4 = -3

Consider the following system of equations.
x1+2x2+2x3 −
2x4+2x5 = 5
−2x1 − 4x3+ x4 −
10x5 = −11
x1+2x2 − x3+3x5 =
4
1. Represent the system as an augmented matrix.
2. Reduce the matrix to row reduced echelon form. (This can be
accomplished by hand or by MATLAB. No need to post code.)
3. Write the set of solutions as a linear combination of vectors
in R5. (This must be accomplished by hand using the rref
form found...

Find the fundamental system of solutions to the system.
2x1 − x2 + 3x3 + 2x4
+ x5 = 0
x1 + 4x2 − x4 + 3x5
= 0
2x1 + 6x2 − x3 + 5x4
= 0
5x1 + 9x2 + 2x3 +
6x4 + 4x5 = 0.

2X1-X2+X3+7X4=0
-1X1-2X2-3X3-11X4=0
-1X1+4X2+3X3+7X4=0
a. Find the reduced row - echelon form of the coefficient
matrix
b. State the solutions for variables X1,X2,X3,X4 (including
parameters s and t)
c. Find two solution vectors u and v such that the solution
space is \
a set of all linear combinations of the form su + tv.

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