MAT 2401Linear Algebra

4.4 II Spanning Sets and Linear Independence

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HW WebAssign 4.4 Part II Written Homework

Review Linear Combination Spanning Set

Review: Example 0 (c)

Preview Linear Independent

Linear Independence

Linear Independence

No one vector is a linear combination of the other vectors.

Example 6Let S={(1,0,0), (0,0,2), (1,0,1)}Determine whether the set S is linearly independent or linearly dependent.

Example 7Let S={(1,1,4), (1,0,3), (1,-1,0)}Determine whether the set S is linearly independent or linearly dependent.

Test

Written HW In this section, two related but

different concepts are introduced: the spanning sets and linear independence.

Written HW Q1 : If S is a spanning set, does it

imply that S is linearly independent?

Written HW Q1 : If S is a spanning set, does it

imply that S is linearly independent?

A1: No.•Construct an counter example in R2 •Find S such that

•S spans R2 and •S is linearly dependent

Written HW Q2 : If S is linearly independent,

does it imply that S is a spanning set?

Written HW Q2 : If S is linearly independent,

does it imply that S is a spanning set?

A2: No.•Construct an counter example in R3 •Find S such that

•S is linearly independent and •S does not span R3