Determinant Linear Independence

Introduction to determinant linear independence:

In mathematics, determinants are linearly independent if none of the determinants can be obtained from the others.
When determinants are linearly independent, then each determinant contains new information about the variables.
For example,
A = `[[0],[0],[2]]` B = `[[2],[-4],[2]]` C = `[[0],[4],[-4]]` D = `[[8],[4],[3]]` .

Here, the determinants A, B and C are linearly independence determinant but the determinant of D is not since determinant of D is equals to 9A + 4B + 5C. Following example and practice problems will help you to study about linearly independence determinant. I like to share this Determinant Calculator with you all through my article.

Example Problems of Linear Independence Determinant:

Example problem 1:

Show that the determinant `[[7,2],[6,4]]` is linearly dependence or not using Wronskian determinant method.

Solution:

Step 1: Given determinant

`[[7,2],[6,4]]` .

Step 2: Condition of Wronskian determinant

If the value of the determinant is equal to 0, then the determinant is linearly dependent.

If the value of the determinant is not equal to 0, then the determinant is linearly independent.

Step 3: Calculate the value of the determinant.

`[[7,2],[6,4]]` = (7 * 4) - (2 * 6)

= 16

Since the value of the determinant is not equal to 0, the given determinant is linearly independence.

Step 4: Solution

Hence, the given determinant `[[7,2],[6,4]]` is linearly independence.

Example problem 2:

Show that the determinant `[[4,2],[2,1]]` is linearly dependence or not using Wronskian determinant method.

Solution:

Step 1: Given determinant

`[[4,2],[2,1]]` .

Step 2: Condition of Wronskian determinant

If the value of the determinant is equal to 0, then the determinant is linearly dependent.

If the value of the determinant is not equal to 0, then the determinant is linearly independent.

Step 3: Calculate the value of the determinant.

`[[4,2],[2,1]]` = (4 * 1) - (2 * 2)

= 0

Since the value of the determinant is equal to 0, the given determinant is linearly dependence.

Step 4: Solution

Hence, the given determinant `[[4,2],[2,1]]` is linearly dependence. Is this topic free math problem solver online hard for you? Watch out for my coming posts.

Practice Problems of Linear Independence Determinant:

1) Show that the determinant `[[6,2],[3,4]]` is linearly dependence or not using Wronskian determinant method.

2) Show that the determinant `[[3,6],[6,12]]` is linearly dependence or not using Wronskian determinant method.

Solutions:

1) The given determinant `[[6,2],[3,4]]` is linearly independence.

2) The given determinant `[[3,6],[6,12]]` is linearly dependence.