basis

eyehategod

Let S be a linearly independent set of vectors from the finite dimensional vector space V. Prove that there exists a basis for V containing S. Can anyone help me out? I can't figure out how to approach this.

CompuChip

What is a basis? What do you need to have one? What do you already have? Can you think of a way to construct the rest?

HallsofIvy

If the independent set already spans the space, you are done. If not then there exists a vector that cannot be written as a linear combination of the vectors in the independent set. Can you show that adding that new vector to the set still gives you an independent set of vectors? If that new set spans the vector space, you are done. If not ....

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eyehategod	basis Tweet Let S be a linearly independent set of vectors from the finite dimensional vector space V. Prove that there exists a basis for V containing S. Can anyone help me out? I can't figure out how to approach this.

CompuChip Blog Entries: 5 Recognitions: Homework Help Science Advisor	What is a basis? What do you need to have one? What do you already have? Can you think of a way to construct the rest?

HallsofIvy Recognitions: Gold Member Science Advisor Staff Emeritus	If the independent set already spans the space, you are done. If not then there exists a vector that cannot be written as a linear combination of the vectors in the independent set. Can you show that adding that new vector to the set still gives you an independent set of vectors? If that new set spans the vector space, you are done. If not ....