Question: Which of the following statements about linear system equations are correct?
Statements:
A non-homogeneous system equations $Ax = b$ with $A$ of size $6\times7$ can have a unique solution for a particular right-hand side $b$.
A homogeneous system equations $Ax = 0$ with the size $6\times6$ matrix $A$ can have the amount of all solutions spanned by two vectors.
A non-homogeneous system equations $Ax = b$ with the size $A$ of size $7\times6$ can have a unique solution for a particular right-hand side $b$.
A system equations $Ax = 0$ with the size $A$ of size $10\times12$ of can have the amount of all solutions consisting of multiples of a vector.
A system equations $Ax = 0$ with the size $7\times10$ matrix $A$ can have the amount of all solutions spanned by two vectors.
My answer:
It stands still in my head and I don't know where to start from to be able control of which statement that is true or false. Please help me!
We can look at it one at a time. I won't just give the answer because it is clearly a homework question but we can work through this together. The first thing you should ask yourself is: What is this question really asking about?
Answer 1 shows an equation
Ax=band asks if A is a 6x7 matrix, can we solve for x. Well, what does a 6x7 matrix mean? How many rows and how many columns? What do the columns mean? Notice there is a mismatch between rows and columns, what does that mean?Answer 2 is like answer one but b = 0, which means what? What does it mean to have a system spanned by vectors?
Answer 3 is like answer 1 but what is different?
Answer 4 is similar to answers 1 and 3 but what is it asking about this time with "multiples of a vector"?
Answer 5 is like answer 2 but what is different?