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.

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(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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Expert Answer

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.

Algebra homework question answer, step 1, image 1
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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,

Algebra homework question answer, step 2, image 1
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Step 3
  1. c) Assume that T is one to one.

Then ...

Algebra homework question answer, step 3, image 1
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