Label the following statements as being true or false. For the following, V and W are finite dimensional vector spaces (over F) and T is a function from V into W. a) If T is linear than 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 N(T)={0}. d) If T is linear then T(0v)=0w. e) If T is linear than 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→ are both linear and agree on a basis of V, then T=U. h) Given x1, x2 ϵ V and y1, y2 ϵ W, there exists a linear transformation T:V→W such that T(x1)=y1 and T(x2)=y2.

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Label the following statements as being true or false. For the following, V and W are finite dimensional vector spaces (over F) and T is a function from V into W.
a) If T is linear than 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 N(T)={0}.
d) If T is linear then T(0v)=0w.
e) If T is linear than 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→ are both linear and agree on a basis of V, then T=U.
h) Given x1, x2 ϵ V and y1, y2 ϵ W, there exists a linear transformation T:V→W such that T(x1)=y1 and T(x2)=y2.

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Step 1

Since there are multiple subparts posted, we will answer the first three subparts (a,b,c).

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Given:

V and W are finite dimensional vector spaces over F and T is a function from V into W.

Step 2
1. b) Given that T(x + y) =T(x) + T(y).

For T to be linear, T must also satisfy the condition T(cx) = cT(x).

Define T as a mapping from C to C by,

Step 3
1. c) Assume that T is one to one.

Then ...

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