Rank of a Matrix and whether the columns span R12

[testme]

Homework Statement

Let M be the 12 x 7 coefficient matrix of a homogeneous linear system, and suppose that this system has the unique solution 0 = (0, ..., 0) $\in$ ℝ⁷.

1. What is the rank of M.
2. Do the columns of M, considered as vectors in ℝ¹², span ℝ¹².

Homework Equations

The Attempt at a Solution

1. Well since the matrix is a homogeneous matrix the rank of M can be from 0 $\leq$ rankM $\leq$ 7.

so then rank has a rank of 7 I believe

2. I'm not sure how to solve this but if the solution is in ℝ⁷ does that automatically mean the vectors can't span ℝ¹² and aside from that a set of all 0s can't span ℝ⁷ can it?

 

[STEMucator]

By 12 x 7 I believe you mean column x row in this case? I ask this because if you had it the other way around, your zero vector would be in ℝ¹² not ℝ⁷.

 

[testme]

No, my professor does it 12 x 7, row x column (I thought that's the norm way of doing it?)

Maybe the ℝ⁷ is a typo by the professor and it should be ℝ¹²

I'm not completely sure.

 

[Mark44]

2. Do the columns of M, considered as vectors in R¹², span R¹².

2. I'm not sure how to solve this but if the solution is in R⁷ does that automatically mean the vectors can't span R¹² and aside from that a set of all 0s can't span R⁷ can it?

This question is asking about the columns of the matrix, not the solutions. The columns are in R¹² and there are seven of them, so could the columns span R¹². Hint: how many vectors does it take to span R³? R³?

 

[testme]

This question is asking about the columns of the matrix, not the solutions. The columns are in R¹² and there are seven of them, so could the columns span R¹². Hint: how many vectors does it take to span R³? R³?

So then if I'm not mistaken since there are only 7 columns (or 7 vectors) and there must be a minimum of n vectors to span ℝⁿ 7 vectors can't span all of ℝ¹². So the answer is no.

As for the first one would I be correct to say that the rank is then 7?