Label the following statements as true or false. In each part, V and W are

QUESTION:

Label the following statements as true or false. In each part, V and W are finite-dimensional vector spaces (over F), and T is a function from V to W.

(a) If T is linear, then T preserves sums and scalar products.

(b) If T(x + y) = T(x) + T(y), then T is linear.

(c) T is one-to-one if and only if the only vector x such that T(x) = 0 is x = 0.

(d) If T is linear, then \(\mathrm{T}\left(0_{\mathrm{V}}\right)=0_{\mathrm{W}}\)

(e) If T is linear, then nullity(T) + rank(T) = dim(W).

(f) If T is linear, then T carries linearly independent subsets of V onto linearly independent subsets of W.

(g) If T, \(U: \mathrm{V} \rightarrow \mathrm{W}\) are both linear and agree on a basis for V, then T = U.

(h) Given \(x_1,\ x_2\in\mathrm{V}\) and \(y_1,\ y_2\in\mathbf{W}\), there exists a linear transformation \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) such that \(\mathrm{T}\left(x_{1}\right)=y_{1}\) and \(\mathrm{T}\left(x_{2}\right)=y_{2}\).