Question

Could a set of three vectors in ℝ^4 span all ℝ^4? Explain. What about n vectors in ℝ^m when n is less than m?

Answer 1

Answer:

Check the ecplanation

Step-by-step explanation:

A set of three vectors in $R^{4}$ represents a matrix of 3 column vectors, and each vector containing 4 entries (that is, a matrix of 4 rows, and 3 columns).

Let A be that 4x 3 matrix. The columns of A span $R^{m}$. if and only if A has a pivot position in each row. So, there are at most 3 pivot positions in the matrix A, but the number of rows is 4, therefore, there exist at least one row not having a pivot position. If A does not have a pivot position in at least one row, then the columns of A do not span $R^{4}$. It implies that the set of 3 vectors of A does not span all of $R^{4}$.

In general, the set of n vectors in $R^{m}$ represents a matrix of in rows, and n columns (an in x matrix). So, there are at most n pivot positions in the matrix A, but n is less than the number of rows. In therefore, there exist at least one row that does not contain a pivot position.

And, hence the set of n vectors of A does not span all of $R^{m}$. for n < m