Mathematics II - Old Questions

16. Let  be a linear transformation and let A be the standard matrix for T. Then prove that: T map Rⁿ on to R^m if and only if the columns of A span R^m; and T is one-to-one if and only if the columns of A are linearly independent. Let T(x₁, x₂) = (3x₁ + x₂, 5x₁ + 7x₂, x₁ + 3x₂). Show that T is a one-to-one linear transformation. Does T map R² onto R³?