Consider linearly independent vectors in a subspace V of and vectors that span V. Show that there is a basis of V that consists of all the and some of the . Hint: Find a basis of the image of the matrix
Hence, a basis of V consists of all the and some of the .
Independent vectors and spanning vectors in a subspace of
Consider a subspace V of with .
a. We can find at most m linearly independent vectors in V.
b. We need at least m vectors to span V.
c. If m vectors in V are linearly independent, then they form a basis of V.
d. If m vectors in V span V, then they form a basis of V.
Removing the dependent vectors from the set of vectors .
Since, span V:
When they are linearly independent then they form a basis of V and when they are linearly dependent then .
Also, there is maximum linearly independent vector in V will be m. Thus, we need to eliminate the redundant vectors of from B
As linearly independent vectors in V form a basis. So, we have to eliminate vectors.
Therefore, we have to eliminate some vectors so that the condition of independent vectors and spanning vectors in a subspace of remains valid.
Hence, proved that, a basis of V consists of all the and some of the
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