**Dot Product**

–

–

**Span**

**Subspaces**

is a subspace of if

1) for some vectors .

2) satisfies the **closure properties**:

(i) for all , we must have .

(ii) for all and , we must have .

3) is the solution set of a homogeneous system.

(Sufficient to check either one of Condition 1, 2, 3.)

Remark:

For to be a subspace, zero vector must be in . (Since for , , we have .)

**Linear Independence and Dependence**

are linearly independent if the system has only the trivial solution, i.e. .

If the system has non-trivial solutions, i.e. at least one not zero, then are linearly dependent.

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