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Vectors as Data
Linear Combination
A linear combination builds a vector by scaling vectors and adding the results.
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Linear Combination
Definition
A linear combination is built by scaling vectors and adding them together.
Example:
2 * [1, 0] + 3 * [0, 1]
First scale:
2 * [1, 0] = [2, 0]
3 * [0, 1] = [0, 3]
Then add:
[2, 0] + [0, 3] = [2, 3]
Linear combination = scaled vectors added together.
Two Operations
Linear combinations use only two operations:
- scalar multiplication
- vector addition
Example:
a * v + b * w
where:
a and b are scalars
v and w are vectors
Linear combinations are built from scaling and adding.
Simple Example
Let:
v = [2, 1]
w = [0, 3]
Build:
4 * v + 2 * w
Step by step:
4 * [2, 1] = [8, 4]
2 * [0, 3] = [0, 6]
[8, 4] + [0, 6] = [8, 10]
Result:
[8, 10]
A linear combination creates a new vector from existing vectors.
Why "Linear"?
It is called linear because the operations are limited.
Allowed:
scale a vector
add vectors
Not part of a linear combination:
square a component
multiply two components together
apply a conditional rule
Those operations can be useful, but they are not linear combination operations.
Linear combinations restrict how vectors can be combined.
Linear Combination As Construction
A vector can be constructed from simpler vectors.
Example:
[5, 2] = 5 * [1, 0] + 2 * [0, 1]
The two simple vectors contribute independently:
[1, 0] controls the first component
[0, 1] controls the second component
Linear combinations show how vectors can be assembled.
Coefficients
The scalars used in a linear combination are called coefficients.
Example:
3 * v + 7 * w
The coefficients are:
3
7
They control how much each vector contributes.
Coefficients are scalar weights in a linear combination.
Why It Matters
Linear combination is a core idea for understanding how vector spaces are built.
It answers a basic question:
What vectors can be created from these starting vectors?
That leads directly to basis vectors, span, and linear independence.
Linear combinations describe what can be built from available vectors.
Final Summary
- A linear combination scales vectors and adds the results.
- It uses scalar multiplication and vector addition.
- The scalars are coefficients.
- It creates new vectors from existing vectors.
- It restricts combination to linear operations.
- It prepares the ground for basis, span, and independence.
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Continue the series.
Basis Vector