# Thread: I am clueless of Span and Linear Transformation.

1. ## I am clueless of Span and Linear Transformation.

I have read over my notes many times but I am still not able to grasp the meaning of span and linear transformation. Can someone please explain them to me?

Thanks

2. A set of vectors span some space if the vectors are linearly independent.

The span doesn't need to span the dimension.

If we are working in $\mathbb{R}^3$, our span could be 2 vectors in the xy plane as long as they aren't linearly dependent.

A linear transformation in V maps the a vector v in V to W following some predefined rule.

L(x) = 3x.

All the vectors from V to W are 3 times longer in W.

3. Originally Posted by Jishent
I have read over my notes many times but I am still not able to grasp the meaning of span and linear transformation. Can someone please explain them to me?

Thanks
Which particular aspect of the terms are you having difficulty? The two concepts are fairly (and I mean this with no disprespect, it's possible your book is just bad at explaining it!) fundamental concepts.

4. There are only two operations in a vector space- adding vectors and multiplying vectors by numbers.

The "span" of a set of vectors is the space of all vectors you can get by adding vectors in the set and multiplying by numbers.

A linear tranformation if a function from one vector space to another (or to itself) that "plays nicely" with those operations: F(u+ v)= F(u)+ F(v) and F(av)= aF(v) where F is the function, u and v are vectors and a is a number.

5. The objects v of linear algebra can be represented as n-tuples of numbers, (x1,x2,x3,..xn), usually called vectors, and a rule for addition of vectors, v1+v2, and multiplication by a scalar. For n=3:

(x1,x2,x3) + (y1,y2,y3) = (x1+y1, x2+y2, x3+y3)
a(x1,x2,x3) = (ax1,ax2,ax3)

A linear transformation is a rule T which assigns to every vector another vector such that
T(x+y)= Tx + Ty
aTx = Tax

Example: assign to every vector v the vector av.

A linear combination of vectors is defined by:
a1v1+a2v2 + a3v3+ ..... anvn.

The span of a set of vectors is all linear combinations of the vectors.

Ex: All linear combinations of two non-parallel vectors in 3d space is a plane.