Linear Algebra Vector-Spaces-3, guaranteed and verified 100% Pass

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Vector Spaces

Vectors in ℝ2

A nonzero vector in ℝ2 can be represented by a directed line segment. So a
vector is something with a magnitude, how long the vector is, and a direction.

Ex. We can think of the vector 𝑣 = < 2, 3 > as a line segment starting at
(0, 0) (or any other point in the plane) and ending 2 units to the right and 3
units up.
(6, 5)
(2, 3)


(4, 2)
(0, 0)



The length of any vector 𝑣 = < 𝑎, 𝑏 > in ℝ2 is |𝑣 | = √𝑎2 + 𝑏 2

Ex. The length of 𝑣 = < 2, 3 > is:
|𝑣 | = √22 + 32 = √4 + 9 = √13

We can multiply any vector in ℝ2 by a real number 𝛼, called a scalar, by
𝑣 = < 𝑎, 𝑏 >
𝛼𝑣 = 𝛼 < 𝑎, 𝑏 > = < 𝛼𝑎, 𝛼𝑏 >

Ex. If 𝑣 = < −3, 2 >
3𝑣 = 3 < −3, 2 > = < −9,6 >
−2𝑣 = −2 < −3, 2 >=< 6, −4 >



If we have 2 vectors:
𝑣 = < 𝑣1 , 𝑣2 >
𝑤 = < 𝑤1 , 𝑤2 >
then 𝑣 + 𝑤 = < 𝑣1 , 𝑣2 > + < 𝑤1 , 𝑤2 > = < 𝑣1 + 𝑤1 , 𝑣2 + 𝑤2 >.

, 2


Geometrically, 𝑣 + 𝑤 is the vector starting at (0,0) and ending at
(𝑣1 + 𝑤1 , 𝑣2 + 𝑤2 ).

(𝑣1 + 𝑤1 , 𝑣2 + 𝑤2 )

𝑣+𝑤
w


(v1, v2)
v




If 𝑣 = < 𝑎, 𝑏 > then −𝑣 = < −𝑎, −𝑏 >.
−𝑣 is the same length as 𝑣 but points in the opposite direction.

Thus 𝑣 + (−𝑣 ) =< 0, 0 >, the
zero vector

v



−𝑣




If w is any vector in ℝ2 then 𝑤+ < 0, 0 > = 𝑤.