Linear Algebra Span of Vectors

[luvlybug1025]

Homework Statement

a)Construct a 3x3 matrix, not in Echelon form, whose columns do NOT span R3. Prove.

b)Can a set of 3 vectors Span all of R4? Explain.

Homework Equations

The Attempt at a Solution

a)OK...I can make up plenty of matrices in Echelon form that fit, but how do I come up with one before it reaches Echelon form? Would it be like:
6 8 -4 3
2 5 1 -1
-2 -5 -1 1
because the last equation will "zero out"?

b)Someone told me this could happen, but I just don't see how. I thought every basis for R4 had to contain exactly 4 vectors?

 

[JG89]

For this question: "b)Can a set of 3 vectors Span all of R4? Explain."

Find a set that spans all of R4, such as {[1,0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1]}

A spanning set is considered minimal (in the sense that if you remove any of the vectors you will change the span and also in the sense that it's the minimum amount of vectors allowed in the spanning set) if none of the vectors in the set can be written as a linear combination of the others in the set, or if the vectors are linearly independent.

This should help you.

 

[Defennder]

a) This isn't too difficult is it? Just find some random vector. Then multiply a constant to it, and you get another vector. Repeat. Now what is the dimension of the subspace spanned by these 3 vectors?

P.S. Your matrix is 4x4, not 3x3 as required.

b) What the largest value the dimension of a subspace spanned by 3 vectors can have? What is the dimension of R4?

 

[HallsofIvy]

Suppose there were 3 vectors, e₁, e₂, e₃, that span R⁴. As JG89 said, [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1] form the standard basis for R⁴ and so, in particular, are in R⁴. If e₁, e₂, and e₃ span the R⁴, those four vector could be written in terms of them. Write out the equations and try to solve for the coefficients.