SOLUTION: Iif there are three linearly independent vectors in a vector space, do the dimensions have to be greater than or equal to three?

Question 925226: Iif there are three linearly independent vectors in a vector space, do the dimensions have to be greater than or equal to three?
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
Three linearly independent vectors form a basis, because any vector in the space can be set as a linear combination of them.
i.e. suppose we have vectors u, v, w then vector x is a combination of them
x = au +bv +cw
The coordinates of the vector that form the base are:
x =(a,b,c) where a,b,c are scalers
now
we can have an Orthogonal basis where the three basis vectors are mutually perpendicular.
we can also have an Orthonormal Basis where the three basis vectors are mutually perpendicular and also have a length of one.
I think you are asking about an Orthonormal Basis:
suppose we have three vectors i, j, k where
i = (1,0,0), j=(0,1,0) and k = (0,0,1) then
|i| = |j| = |k| = 1
i is perpendicular to j, j is perpendicular to k, k is perpendicular to i
note that a vector has direction and magnitude, for example
magnitude of i is written |i| = square root(1^2 + 0^2 + 0^2) = 1
Then every vector a in three dimensions is a linear combination of the orthonormal basis vectors i, j, and k.
For example, every vector a in three-dimensional space can be written uniquely as a1i + a2j +a3k
where a1, a2, a3 being the scalar components of the vector a.
The dimension of a vector space V is the number of vectors of a basis of V, in your example the dimension is equal to 3.

RELATED QUESTIONS
proofs kill me! S=(v1,v2,...vn) is a linearly independent set of vectors in the vector... (answered by venugopalramana)

Let x and y be vectors in R^4 with {x,y} being linearly independent. Prove that there... (answered by richard1234)

Let V be a vector space with dimension 11 and S be a subset of V which is linearly... (answered by ikleyn)

Am I allowed to simple ask for assistance into understanding the concept of vectors in... (answered by rothauserc)

Let b1, b2, b3 be 3 vectors over Z5 with dimensions 3x1. b1, b2, b3 are linearly... (answered by ikleyn)

vector A=3i+4j is a vector in xy plane, vector B is a vector perpendicular to vector A, (answered by longjonsilver,venugopalramana)

Let v1= [0, -9, -9], v2 = [1, -3, -3], v3 = [1, 0, 0] Determine whether or not the... (answered by ikleyn)

If {vector u, vector v, vector w} is linearly independent then u, v, and w are not in R^2 (answered by Fombitz)