Final answer:
To find the value(s) of h for which y is in the span of V₁, V₂, and V₃, we express y as a linear combination of V₁, V₂, and V₃ and solve for the coefficients. this leads to a system of linear equations that can be solved to determine the values of h that satisfy the condition.
Step-by-step explanation:
To determine for what values of h the vector y will be in the Span{V₁, V₂, V₃} we need to set up a system of linear equations where y can be expressed as a combination of V₁, V₂, and V₃:
y = c₁V₁ + c₂V₂ + c₃V₃
Looking at the vectors, and noting that there is a typo (V₁ is listed twice with different values, we assume the third vector should be named V₃), V₁ = [1 -1 -2], V₂ = [5 -4 -7], V₃ = [-3 1 4], and y = [-4 3 h], we set up the following matrix equation:
[ 1 5 -3] [c₁] = [-4]
[-1 -4 1] [c₂] = [ 3]
[-2 -7 4] [c₃] = [ h]
To solve for c₁, c₂, and c₃, we need to use row reduction or another method of solving systems of linear equations. After finding the values of c₁, c₂, and c₃, we can determine the value of h that allows y to be expressed as a linear combination of V₁, V₂, and V₃. This value or values of h will satisfy the condition that y is in the span of V₁, V₂ and V₃.