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AlgebraQ&A LibraryLabel 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.Question

Asked Jan 14, 2020

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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(0_{v})=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 x_{1}, x_{2} ϵ V and y_{1}, y_{2} ϵ W, there exists a linear transformation T:V→W such that T(x_{1})=y_{1} and T(x_{2})=y_{2}.

Step 1

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

If you want any specific subpart to be answered, then resubmit the question and specify the subpart you want us to answer.

Given:

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

Step 2

- b) Given that T(
*x*+*y*) =T(*x*) + T(*y*).

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

Define T as a mapping from **C** to **C** by,

Step 3

- c) Assume that T is one to one.

Then ...

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