What Is the Linear Property of Determinants?

[Winzer]

So I am looking at the proof for this in a linear algebra book and I half way get it:

Theorem:

If the all elements of the jth column of a determinate D are linear combinations of two columns of numbers, i.e., if
$$D=\lambda b_{i}+uc{i}$$ where lambda and mu are fixed numbers, then D is equalto a linear comination of the two determinates:
$$D=D_{1}\lambda+D{2}u$$

Here both determinates D1 and D2 have the same columns as the determinate D except for the jth column; the jth colum of D1 consists of the numbers $$b_{i}$$ wile the jth column of D2 consists of the numbers $$c_{i}$$

 

[Winzer]

Actually I figured it out, sorry for the dumb question.
I have never dealt with any proofs before but I get it.

 

[tacman]

The linear property of determinates essentially states that if we have a determinate where the elements of one column are a linear combination of two other columns, then the value of that determinate can be expressed as a linear combination of the two determinates formed from the two columns. This means that the value of the determinate is dependent on the values of the individual determinates formed from the two columns.

In other words, if we know the values of the determinates formed from the two columns, we can use them to find the value of the original determinate. This property is important because it allows us to manipulate determinates in a linear manner, making calculations and proofs much easier.

The proof for this theorem involves using cofactor expansion and properties of determinates, and it can be a bit complex. However, the key idea is that by rearranging the columns and using properties of determinates, we can show that the value of the original determinate is equal to a linear combination of the two determinates formed from the two columns.

Overall, the linear property of determinates is a fundamental concept in linear algebra, and it has many applications in areas such as matrix operations, solving systems of linear equations, and finding areas and volumes in geometry. Understanding this property is crucial in fully grasping the concept of determinates and their role in linear algebra.