Linear Combinations and Spans

Author:Ku, Yin Bon (Albert)

Linear Combinations and Spans

We can easily extend our define of linear combination and span to general vector spaces: Definition: For vectors

and real numbers

is called a linear combination of

with weights numbers

Definition:

i.e. the set of all linear combinations of

. Remark: More generally, for any set

of vectors in V (possibly infinite set),

is the set of all linear combinations of any finite number of vectors in

. Theorem:

is a subspace of

. Proof: We just need to verify the three conditions:

The zero vector is a linear combination of .
If and are linear combinations of , then so is .
If is any real number and is a linear combination of , so is .

Also, the definition of linear independence is essentially the same as before: Definition: The set

is said to be linearly independent if

implies

. The set is said to be linearly dependent if it is not linearly independent. Remark: More generally, for any set

of vectors in

(possibly infinite set),

is said to be linearly independent if any finite subset of

is linearly independent. The following is an equivalent definition of linearly dependence: Theorem: A set of vectors

in a vector space

is linearly dependent if and only if some

is a linear combination of the others. We already proved the theorem for

. The same proof also works for general vector spaces. Example 1: Consider the set of polynomials

. Show that

is a linearly independent set and find

. Linear independence: Suppose

are real numbers such that

. It is obvious that

. Hence

is a linear independent set.

Example 2: Consider the set

. Is the set linearly independent? To check whether the set is linearly independent, we need to consider the following equation:

And it is equivalent to solving the linear system

. It is not hard to see that the system is consistent and has nontrivial solutions like

. That is to say, the weights of the linear combination are not all zero, which implies that the set is linearly dependent.

Exercise

Is the set in a linearly independent set? Explain your answer briefly.

Check my answer

In , let be the sequence , be the sequence and so on. What is ?

Check my answer

Linear Combinations and Spans

Linear Combinations and Spans

Exercise

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