Find a basis of a linear space and thus determine its dimension. Examine whether a subset of a linear space is a subspace. Which of the subsets of given in Exercises 1 through 5 are subspaces of (see Example 16)? Find a basis for those that are subspace,
is a subset and a subspace of .
A subset W of a linear space V is called a subspace of V if
(a) W contains the neutral element 0 of V.
(b) W is closed under addition (if f and g are in W then so is f+g )
(c) W is closed under scalar multiplication (if f is in W and k is scalar, then kf is in W ).
we can summarize parts b and c by saying that W is closed under linear combinations.
General expression for a quadratic polynomial with one variable is
Also, let us assume two scalars ,
Then, we can write as:
Therefore, the given set is closed under linear combinations. Hence, it is a subspace of .
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