4
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Let $A$ be a $5×4$ matrix with real enries such that the space of all solutions of the linear system $AX^t=[1,2,3,4,5]^t$ is given by $\{[1+2s,2+3s,3+4s,4+5s]^t:s\in\mathbb{R}\}$. Then the rank of $A$ is equal to

  • $4$
  • $3$
  • $2$
  • $1$

I am completely stuck on it. Can anyone help me please.

  • 3
    Do you know the [Rank-Nullity Theorem](http://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem)?2012-12-16
  • 0
    yes .but how can i apply this?2012-12-16
  • 0
    Apply it to the matrix $A$. You want to solve for the rank of $A$, and you know the number of columns of $A$. So all that remains is to calculate the nullity of $A$. Can you do that given the information about the linear system? Another way of asking this is: what is the connection between the nullity of a matrix and the solution space of a linear system with that coefficient matrix?2012-12-16
  • 0
    You have $A:\mathbb{R^4}\rightarrow \mathbb{R^5}$ and $X\in \mathbb{R^5}$, so how you can apply $A$ to $X$?2012-12-18

1 Answers 1