Linear Algebra Example Problems - Linear Combination of Vectors #1

Given the vectors v1, v2, and v3, we see if the vector b can be written as a linear combination of the vectors.

This can be easily determined by constructing an augmented matrix, performing row operations, and finding the coefficients such that a1*v1 + a2*v2 + a3*v3 = b.

If values for a1, a2, and a3 can be found, then b is a linear combination of {v1,v2,v3} and we say that b is in the Span{v1,v2,v3}. If the augmented matrix has no solution, then b is NOT a linear combination of the vectors.

For this example, b CAN be written as a linear combination.

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[TheOnlyRealBreadIntheWorld]

Thank you so much. You are a lifesaver. The world always needs great teachers like you

[eldiospadre]

Short, to the point, well-explained. Excellent video!

[janakicharles]

This is such a brilliant way of explaining this topic.

Thanks. Your explanation is very clear.

[ED-ixmq]

Awesome explanation, thank you very much!

[meenamena]

Express the vector (3, 5, 2) as a linear combination of the vectors (1, 1, 0, ) (2, 3, 0) and (0, 0, 1) of v3 (R) pz solved madi kodi

[petitbonheur]

But is b still a linear combination of these vectors if let's say one of the columns is a free variable column, we just have to write our alphas in parametric form right???

[Herotruth]

Why didn't you show one that was not so we can see the difference?

[EmapMe]

Did you really have to solve this using a matrix? Wouldnt it be way easier to just solve it normally?