I have a question that says...

Decide whether the following sets of vectors give a basis for the indicated space. ${(1,0,2,3), (0,1,1,1), (1,1,4,4)};\mathbb{R}^4$

I know a set of vectors is called a basis if...

1)$V_1$,...$V_k$ span V, that is V=Span($v_1$,...$V_k$ )

2)${V_1,...V_k}$ is linearly independent

I have found that the set of vectors are linearly independent by making the matrix and then reducing it. However, I'm confused how I'm suppose to show/check the first part,

1)$V_1$,...$V_k$ span V, that is V=Span($V_1$,...$V_k$ )

Can someone explain how you would do this, so that I can determine if the vectors give a basis.

  • $\begingroup$ Are you asking if the three vectors form a basis for $\mathbb{R}^4$? $\endgroup$ – copper.hat Dec 3 '16 at 19:07
  • $\begingroup$ If $B_1, B_2$ are bases for a given space, what can we say about their cardinality? $\endgroup$ – copper.hat Dec 3 '16 at 19:12

Three vectors can never give a basis for $R^4$. All bases for $R^4$ will have 4 elements-in fact, it is a necessary and sufficient condition for m vectors to be a basis of $R^n$ that $m=n$ and all of the vectors are linearly independent.


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