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- 1,753

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http://math.uh.edu/~bgb/Courses/Math4377/Math4377-Ex1-Topics.pdf

1.2

**Does the solution set to a linear system change under elementary row operations?**

The solution set does not change under elementary row operations.

**What are independent/free variables? How can we tell there are free variables by looking at the row-reduced echelon form?**

I would give an example:

1 0 0 0

0 1 0 0

0 0 0 0

x1=x2=independent, x3=x4=free=linear combination of alpha1 (0 1 0 0) & alpha2 (0 0 0 0)

1.3

**How can we rewrite a linear system Ax = b in vector form?**

Ax=b

A=

A11 ... A1n

.

.

.

Am1 ... Amn

x=

x1

.

.

.

xn

b=

b1

.

.

.

bm

**Can we solve the system if b can be written as a linear combination of the column vectors of A?**

Yes, I would just put it in augmented form and apply elementary row reductions. I would find that some columns have no pivot variables, thus the variable in that column is free, which is the consequence of being a linear combination of other columns.

1.4

**How do the solutions to an inhomogeneous system relate to the solutions of the corresponding homogeneous one?**

Homogeneous systems are equal to zero a1x1+...anxn=0, thus it is linear independent in which the a1=...=an=0 or it contains only the trivial solution, all xn=0.

Inhomogeneous systems are not equal to zero, thus it's solutions will not all be zero.

THANKS!!!