Consider the plane (x,y,z)=(0,1,1)+λ(1,-1,1)+m(2,-1,0). The distance of this plane from the origin i

Tapas Gandhi

Consider the plane (x,y,z)=(0,1,1)+λ(1,-1,1)+m(2,-1,0). The distance of this plane from the origin is?How to read the Linear Combination of Vectors?

Deepak Aggarwal

Linear Combination of Vectors

A linear combination of vectors is a sum of scalar multiples of those vectors. That is, given a set of vectors $underline{x}_i$ of the same type, such as ${f R}^N$ (they must have the same number of elements so they can be added), a linear combination is formed by multiplying each vector by a scalar and summing to produce a new vector of the same type:

$displaystyle underline{y}= alpha_1 underline{x}_1 + alpha_2underline{x}_2 + cdots + alpha_M underline{x}_M$

For example, let $underline{x}_1=(1,2,3)$ , $underline{x}_2=(4,5,6)$ , , and . Then the linear combination of $underline{x}_1$ and $underline{x}_2$ is given by

$displaystyle underline{y}= alpha_1underline{x}_1 +alpha_2underline{x}_2 = 2cdot(1,2,3) + 3cdot(4,5,6)= (2,4,6)+(12,15,18) = (14,19,24).$

In signal processing, we think of a linear combination as a signal mix. Thus, the output of a mixing console may be regarded as a linear combination of the input signal tracks.

Source: https://ccrma.stanford.edu/~jos/st/Linear_Combination_Vectors.html

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Linear Combination of Vectors