8.4 Column Space and Null Space of a Matrix
... Note that the left side of this equation is simply a linear combination of the columns of A, with the scalars being the components of x. The system will have a solution if, and only if, b can be written as a linear combination of the columns of A. Stated another way, we have the following: Theorem 8 ...
... Note that the left side of this equation is simply a linear combination of the columns of A, with the scalars being the components of x. The system will have a solution if, and only if, b can be written as a linear combination of the columns of A. Stated another way, we have the following: Theorem 8 ...
Homework # 2 Solutions
... Solution: For any vectors u, v ∈ Rn , T (u + v) = A(u + v) + b T (u) + T (v) = Au + b + Av + b = A(u + v) + 2b 6= T (u + v) unless b = 0. Therefore, T is not a linear transformation when b 6= 0. Alternate solution: T (0) = A(0) + b = b 6= 0, so T is not a linear transformation when b 6= 0. 36. Let T ...
... Solution: For any vectors u, v ∈ Rn , T (u + v) = A(u + v) + b T (u) + T (v) = Au + b + Av + b = A(u + v) + 2b 6= T (u + v) unless b = 0. Therefore, T is not a linear transformation when b 6= 0. Alternate solution: T (0) = A(0) + b = b 6= 0, so T is not a linear transformation when b 6= 0. 36. Let T ...
