A linear combination is essentially a way to combine vectors or matrices using scalar multiplication and addition.
Definition of Linear Combination
According to the formal definition, a linear combination involves taking a set of matrices (or vectors), multiplying each of them by a scalar (a number), and then adding all the resulting products together.
Breaking Down the Concept
Here’s a simpler explanation:
- Start with a set of vectors or matrices: These are the building blocks.
- Multiply each element by a scalar: A scalar is just a number. Each vector or matrix gets multiplied by its own scalar.
- Add the scaled elements together: This is where you add up all the results from step 2. The final result is the linear combination.
Example
Imagine you have two vectors:
- v = [1, 2]
- w = [3, 4]
You can create a linear combination of these vectors. Let's say we choose the following scalars:
- a = 2 (scalar for v)
- b = 3 (scalar for w)
The linear combination would be:
2v + 3w = 2[1, 2] + 3[3, 4] = [2, 4] + [9, 12] = [11, 16]
So, [11, 16] is a linear combination of v and w.
Key Points
- The scalars can be any real numbers (positive, negative, or zero).
- The result of a linear combination is always an object of the same type as the original vectors or matrices.
- Linear combinations are fundamental in linear algebra, used in concepts like span, basis, and solving systems of linear equations.
Practical Insights
- Span: The set of all possible linear combinations of a set of vectors defines its span.
- Linear Independence: Vectors are linearly independent if no vector in the set can be written as a linear combination of the others.
- Basis: A basis for a vector space is a set of linearly independent vectors that span the entire space.
