Understanding Linear Combinations of Vectors

[Bashyboy]

Hello Everyone,

Pardon me if the following is incoherent. From what I understood of what my professor said, he was basically saying that when a vector can be written as a linear combination of some vectors in a span, this means, geometrically, that the vector is in the plane that the span of vectors defines. How true is this? And if it is so, does anyone know of a good example that illustrates this point?

Thank you.

 

[mathman]

To clarify, a vector written as a linear combination of two vectors is in the plane defined by the two vectors. If the span has more than two vectors, it could define a higher dimensional object.

 

[HallsofIvy]

I'm not sure that "plane" is the right word since that tends to imply two dimensions. A more general term would be "hyper-plane" but even that tends to imply that we are working in Rⁿ. The most correct word is "subspace".

Consider the vectors <3, 1, 1, 0>, <2, 2, 0, 4>, <5, 3, 1, 4>, and <1, 0, 1, 0> in R⁴.
By definition of 'span' any vector in the span of those four vectors can be written as a linear combination, a<3, 1, 1, 0>+ b<2, 2, 0, 4>+ c<5, 3, 1, 4>+ d<1, 0, 1, 0>.

But those four vectors are NOT "independent". In particular, <5, 3, 1, 4>= <3, 1, 1, 0>+ <2, 2, 0, 4> so that is the same as (a+ c)<3, 1, 1, 0>+ (b+ c)<2, 2, 0, 4>+ d<1, 0, 1, 0>. That is a 3 dimensional hyper-plane (subspace) of R⁴.

For another example, let $f(x)= x^3+x^2+ 2x$, $g=x^2+ 3x- 4$, and $h(x)= x^3+ 2x^2+ 5x- 4$ in the vector space of cubic polynomials in x (which is four dimensional). Any vector in their span can be written as $a(x^3+x^2+ 2x)+ b(x^2+ 3x- 4)+ c(x^3+ 2x^2+ 5x- 4)$. But $x^2+ 2x^2+ 5x- 4= (x^3+ x^2+ 2x)+ (x^2+ 3x- 4)$ so that is $(a+c)(x^3+x^2+2x)+(b+c)(x^2+ + 3x- 4)$. That is a two dimensional subspace but I wouldn't call it either a "plane" or a "hyper-plane".