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 T(0 V) = 0 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: V → W are both linear and agree on a basis for V, then T = U. (h) Given x1, x2∈ V and y1, y2∈ W, there exists a linear transformation T: V → W such that T(x1) = y1 and T(x2) = y2.
Label the following statements as true or false. In each part, V and W are finite-dimensional
(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 T(0 V) = 0 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: V → W are both linear and agree on a basis for V, then T = U.
(h) Given x1, x2∈ V and y1, y2∈ W, there exists a linear transformation T: V → W such that T(x1) = y1 and T(x2) = y2.
Trending now
This is a popular solution!
Step by step
Solved in 8 steps with 7 images