Question

Consider the following three vectors of {R⁴ : vec{a₁}=[-3,-3,-5,4], vec{a₂}=[0,-2,0,2], vec{a₃}=[0,0,4,4] There exists a linear equation in the coordinates[x,y,z,u] whose solution set coincides with span{a₁, a₂, a₃}. Determine such a equation (recall an equation must contain an = sign)

Answer 1

To find a linear equation corresponding to the span of the vectors ℓ{a_1}, ℓ{a_2}, ℓ{a_3}, we seek a vector orthogonal to these vectors. The components of this vector would form the coefficients in the linear equation.

Step-by-step explanation:

The student is asking to determine a linear equation whose solution set coincides with the span of the given vectors ℓ{a_1}, ℓ{a_2}, ℓ{a_3} in {R}^4.

The span of these vectors is a subspace of R^4, and since we have three vectors, this subspace can at most be 3-dimensional, meaning there is at least one linearly independent equation that these vectors satisfy. By setting up a matrix with ℓ{a_1}, ℓ{a_2}, ℓ{a_3} as its rows and performing row reduction, we could find a relationship among the coordinates x, y, z, u which corresponds to this subspace.

However, since a direct method of finding such an equation is not provided, we need to understand that this task implies finding a vector that is orthogonal to the given vectors and writing an equation that corresponds to all scalar multiples of it. The coefficients of this linear equation would correspond to the components of the orthogonal vector.