Question

Linear Algebra-- Subspaces of Vector Spaces Determine whether the set W is a subspace of R^3...

Linear Algebra-- Subspaces of Vector Spaces

Determine whether the set W is a subspace of R^3 with the standard operations. Justify your answer.

(a): W={(0,x2,x3): x2 and x3 are real numbers}

(b): W={(a, a-3b, b): a and b are real numbers}

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Answer #1

Solution:

is nonempty.

Let    where   are real numbers

, since   are real numbers .

is closed under addition.

Let   .

Then   , since are real numbers .

is closed under scalar multiplication.

Hence,   is a subspace of .

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is a nonempty.

Let   ,   where are real numbers.

, since   are real numbers.

is closed under addition.

Let  .

Then   , since   are real numbers.

is closed under scalar multiplication.

Hence,   is a subspace of .

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