# Free columns-combinations of columns?

• sarvesh0303
In summary, Gilbert Strang discusses pivot columns and free columns in his book on linear algebra. Earlier columns refer to columns that are to the left of each other. Pivot variables and free variables have different properties, with pivot variables being chosen first and values for free variables being chosen afterwards to find special solutions. However, this method does not always work, so starting with the free variables always produces a solution. This is because what is done to the free variables also affects the pivot variables. On a deeper level, starting with a certain variable is equivalent to letting it be a free variable. This can be seen by interchanging columns to make a different variable pivot. This understanding can greatly clarify the concepts for the reader.
sarvesh0303
In Gilbert Strang's book on linear algebra, he mentions that pivot columns are not combinations of earlier columns and that free columns are combinations of earlier columns. What columns is he referring to by earlier columns?

Also, what property distinguishes pivot variables and free variables that we choose the values for free variables for special solutions. Why don't we choose the values for the pivot variables and then find the free variables?

A is earlier than B means A is to the left of B.

Because it does not always work, starting with the pivot variables. But starting with the free variables always works.

For example, let's say that after Gaussian elimination, you end up with x+y=6, and no equations at all for z and w. X is the pivot variable. y, z and w free. You could start with x, and say y is 6-x, but then what do you do about z and w? But if you do it the other way, then what you do to the free variables tells you all about the pivot variables, too.

On a deeper level, "starting with x" is another way of saying "let x be a free variable". This would correspond to interchanging the columns for x and y to make y pivot and x free.

Thanks a ton. Wow... you just made things so much clearer for me!

## 1. What are free columns-combinations of columns?

Free columns-combinations of columns refer to a type of column arrangement in which individual columns are not connected or restrained to each other. This allows for more flexibility in design and load distribution.

## 2. What are the benefits of using free columns-combinations of columns?

One of the main benefits of using free columns-combinations of columns is the ability to create unique and visually appealing structures. It also allows for easier construction and maintenance, as each column can be individually replaced if needed.

## 3. What are the different types of free columns-combinations of columns?

Some common types of free columns-combinations of columns include portal frames, cantilever frames, and braced frames. Each type has its own advantages and is suitable for different types of structures.

## 4. Are there any limitations to using free columns-combinations of columns?

While free columns-combinations of columns offer many benefits, they also have some limitations. These include reduced stiffness and stability compared to more traditional column arrangements, and the need for more careful consideration of load distribution.

## 5. How do free columns-combinations of columns compare to other column arrangements in terms of cost?

The cost of using free columns-combinations of columns can vary depending on the specific design and materials used. In some cases, it may be more cost-effective than traditional column arrangements, while in others it may be more expensive.

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