# What Is A Free Variable In Linear Algebra, Parameter Vs

What do the leading variables and free variables in a matrix mean? I have the system below and am trying to understand which are which. I searched a lot for this, please help me ! \$\$w + x + y + z = 6 qquad w + y + z = 4 qquad w + y = 2\$\$

The terms “leading variable” and “free variable” are usually defined for the matrix representing a system, and only when the matrix is in row-echelon form.

The augmented matrix for your system is \$\$pmatrix{1&1&1&1&|&6cr1&0&1&1&|&4cr1&0&1&0&|&2cr}\$\$ Notice that each column corresponds to a variable.

Do you know how to use elementary row operations to bring this matrix to row-echelon form?

In row-echelon form, the first non-zero entry in each row (if there is one) is a \$1\$, and the column it is in corresponds to a leading variable; the columns that don”t have that kind of \$1\$ correspond to the free variables. Essentially, columns that don”t have a leading variable, have a free variable.

So for example if a row-echelon form is given by \$\$pmatrix{1&2&3&4&5&|&6cr0&0&1&7&8&|&9cr0&0&0&1&pi&|&sqrt2cr}\$\$ then the 1st, 3rd, and 4th variables are leading variables while the 2nd and 5th variables are free variables.

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edited Sep 1 “15 at 14:20
user174622
answered Mar 21 “14 at 12:14

Gerry MyersonGerry Myerson
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If a set of linear equations can be expressed as let”s say

a = 3x + 4y + 5z – 12

b = 2x + 8y + z – 11

c = 9x + 7y -z – 15

where

The left hand variables don”t appear on the right side and vice versa.On the left side, there is only one variable.

You are watching: What is a free variable in linear algebra

Then,

x, y and z can take values of any combination and are called free variables.

a, b and c are dependent on the above free variables (x, y and z) and cannot be any combination. a, b and c are called pivot or leading variables.

One more thing, depending upon how we form the above equations the pivot and free variable might be changed. For ex.,

The same equation can be expressed as

I thought of explaining the free/leading variables in a non-matrix way, since the original query was in a non-matrix way. Thanks to Gilbert Strang for getting me again started with Linear Algebra.

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answered Sep 1 “16 at 4:50

Praveen SripatiPraveen Sripati
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