Plesae help with thia linear
algebra question
Defination : A subset S of V is said to be a basis of v if and only if S is a linearly independen set and S spans V .
Now V is a vector space over a field K and S is a linearly independent subset of V .
If span (S) = V then by the above defination S forms a basis of V and so S contains in a basis is true .
But if span (S) V , then there
exist v1
V\ S such that
S1 = S
{ v1 }
is a linearly independent set .
If span ( S1 ) = V then S1 forms a basis of v and S1 contains S so S contains in a basis of V .
But if span (S1) V then there exist
v2
V\ S1
such that S1 = S
{ v1 ,
v2 } is a linearly independent set .
If span ( S2 ) = V then S2 forms a basis of v and S2 contains S so S contains in a basis of V .
Proceeding this way we can find a basis of V such that S contained in that basis .
N.B. This is so called EXTENSION LEMMA .
.
.
If you have any doubt please comment .
Plesae help with thia linear algebra question (20) Let V be a vector space over the...
Please help me with this Linear
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(22) Prove that if V is a vector space of dimension n, and that if S is a linearly independent subset of S of cardinality n, then S is a basis of V
Problem 9. Let V be a vector space over a field F (a) The empty set is a subset of V. Is a subspace of V? Is linearly dependent or independent? Prove your claims. (b) Prove that the set Z O is a subspace of V. Find a basis for Z and the dimension of Z (c) Prove that there is a unique linear map T: Z → Z. Find the matrix representing this linear map and the determinant of...
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Let T: V-W be a linear transformation between vector spaces V and W (1) Prove that if T is injective (one-to-one) and {vi,.. ., vm) is a linearly independent subset of V the n {T(6),…,T(ền)} is a linearly independent subset of W (2) Prove that if the image of any linearly independent subset of V is linearly independent then Tis injective. (3) Suppose that {b1,... bkbk+1,. . . ,b,) is a...
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linear algebra help
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How do I do these linear algebra questions?
The question is:
Consider the Vector Space V and its subset W given below.
Determine whether W forms a subspace of V. If your answer is
negative then you must provide which subspace requirement is
violated.
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linear independence question
20. Let V1, V2, ...,Vn be linearly independent vectors in a vector space V. Show that V2,...,Vn cannot span V.
I need the answer to problem 6
Clear and step by step please
Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations 1. Show that. TU is also a linear transformation. 2. Show that aT is a linear transformation for any scalar a. 3. Suppose that T is invertible. Show that T-1 is also a linear transformation. Problem 5. Let T : R3 →...