# 3x5 Matrix whose column span is R3

Hello everyone, I would like to confirm my understanding of these two problems because I believe I have the answer, but I may be incorrect.

For the first question, I know that there can be at most 3 linearly independent vectors within A, which would cause the solution space to be at most 2, or nullity = 2. Given that fact, The statement that there could be 3 linearly independent solutions to Ax = 0 is false.

For the second question, the solution space could either be nothing (no solution exists) because the matrix could be filled with 0's, or infinite because of there are more columns than rows, indicating at least one free variable.

I would appreciate any feedback!

You’re correct about the first part. To span $$\mathbb{R}^3$$ you need three pivot columns, and to have a linearly independent set of three vectors that solve $$Ax = 0$$ you need three columns that are not pivot columns. Since we have five columns, only one of these possibilities can happen.

For the second question, notice that you’re being asked for the shape of the solution set. There are four possibilities:

• the empty set (which you identified)
• a single point
• a line
• or the entire plane.

Which of these are possible?

You should give an example for the cases that are possible.

• Your remark about the shape of the solution set makes a lot of sense, thank you! I was having trouble understanding what the question meant. I believe a line is possible, for example a matrix that has two linear independent vectors, <1,0>, <0,1>, and its combination of <1,1> and then an arbitrary matrix 2x1 matrix, say <3,2>. The free variable that I get from solving this equation will allow me to construct a line I believe, and a plane is possible with 1 linear independent vector, and two free variables. Is this the right thought process? – Trebond Mar 15 at 5:18