Linear combinations

A linear combination is a sum of scalar multiples of equal-dimension vectors.

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On this page 7 sections

Overview
A linear combination builds a vector by scaling supplied vectors and adding the results
Weights can be positive, negative, or zero
For u=(1,1) and v=(1,-1), join 2u and v head to tail
Key takeaway
Sources & further reading
Concept connections

01A linear combination builds a vector by scaling supplied vectors and adding the results#

A linear combination builds a vector by scaling supplied vectors and adding the results. In $au+bv$ , the numbers $a$ and $b$ are the weights; each weight multiplies every coordinate of its vector.

With $u=(1,2)$ and $v=(3,-1)$ , find $2u-v$ . Scaling gives $(2,4)$ and $(-3,1)$ . Their first coordinates sum to −1; their second coordinates sum to 5.

2u-v=(-1,5)

Check your reasoning

For $u=(2,1)$ and $v=(1,-2)$ , find $u+2v$ .

A $(3,-1)$
B $(4,-3)$
C $(6,-2)$

Show answer and explanation

$(4,-3)$

Double $v$ to $(2,-4)$ , then add $u$ : $(4,-3)$ .

02Weights can be positive, negative, or zero#

Weights can be positive, negative, or zero. A zero weight contributes a zero vector. The weights do not have to add to one. With more vectors, scale each by its own weight and add all the contributions.

Check your reasoning

For $u=(4,2)$ and $v=(3,-1)$ , find $0u-2v$ .

A $(-6,2)$
B $(6,-2)$
C $(-2,4)$

Show answer and explanation

$(-6,2)$

Zero removes $u$ . Scale both entries of $v$ by −2.

03For u=(1,1) and v=(1,-1), join 2u and v head to tail#

For $u=(1,1)$ and $v=(1,-1)$ , join $2u$ and $v$ head to tail. The total is $(3,1)$ .

Figure 1Equal-scale axes use 40 pixels per unit in both directions. With u=(1,1) and v=(1,-1), the arrow 2u goes from (0,0) to (2,2). A translated copy of v continues to (3,1). The resultant 2u+v goes from (0,0) to (3,1). The weights 2 and 1 sum to 3.

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Check your reasoning

A state gets update $(1,3)$ twice, then reverses update $(4,1)$ . Total change?

A $(6,7)$
B $(-6,4)$
C $(-2,5)$

Show answer and explanation

$(-2,5)$

Combine $(2,6)$ with $(-4,-1)$ to get $(-2,5)$ .

Key takeaway

Scale each whole vector by its weight, then add matching coordinates.

Construct a weighted sum of supplied vectors.

Sources & further reading

[1]
Ohio State Ximera: Linear combinations and linear independence ↗Ohio State Ximera · Book

01A linear combination builds a vector by scaling supplied vectors and adding the results#

02Weights can be positive, negative, or zero#

03For u=(1,1) and v=(1,-1), join 2u and v head to tail#

Scale each whole vector by its weight, then add matching coordinates.

Sources & further reading

Reference this concept