1.3 Vector Equations.

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Presentation transcript:

1.3 Vector Equations

Geometric Interpretation? Vectors in R2 : 1-column matrices or ordered pairs of real numbers Example: Given and , find Geometric Interpretation?

Vectors in R3 : column matrices or points in three-dimensional space Vectors in Rn : column matrices

Two vectors are equal if and only if their corresponding entries are equal. The vector (where c is a real number) is a scalar multiple of .

Algebraic Properties of

Linear Combinations Given vectors and given scalars is a linear combination of with weights Example:

Example: Determine whether w can be generated as a linear Combination of v and v , where , , and .

A vector equation has the same solution set as the linear system whose augmented matrix is can be generated by a linear combination of vectors in if and only if the following linear system is consistent:

Definition If , then the set of all linear combinations of is denoted by Span and is called the subset of spanned by

Geometric Description of Span: